Thank you. Well, first of all, thanks to the organizers for the invitation to be here. It's really nice to be in the video. So I probably, many of you have already heard me speak about these things. So you will probably be bored. I'm sorry. But basically what I will speak about is a couple of works with Patrice Le Calvé and Maisam Nasiri, which we are finishing the second part, like the first part was quite a few years ago. And the second part kind of completes, in a sense, the picture. So I'll try to give the whole picture this time. So the general problem that we are interested in is the following. We have a surface orientable and a homeomorphism, which preserves orientation. And we have an open connected set, which is invariant by the dynamics. And we assume some kind of, some nice property of this open set. For instance, that it has finitely many topological ends or something like this. But for the, usually the problem reduces to this simpler setting that will be the main focus, which is, we assume that the open set is simply connected. It's much easier to consider simply connected set. And the more general situations can be reduced to this case. So we will consider this case mostly. So the question is, we want to study the dynamics in the boundary of these kind of sets. So the dynamics in the boundary and the relationship between the topology of the boundary and the dynamical restrictions that you can have because of this. The idea is that the boundary of an open set of this kind is kind of like a one-dimensional object in a way. But on the other hand, as I will show later, it can be really bad topologically. So it could contain very complicated dynamics. So there is an interplay between the topology and the dynamics. The motivation for this problem comes in part from some problems in CR generic dynamics of area-preserving defiomorphism. For instance, when you have, I have a picture here. The typical picture that you see when you take an area-preserving defiomorphism is something like this. You have these invariant islands, which are produced by KAM phenomenon.  And then you have, in the complement of these islands, you have this region, which is called the instability region. Some region where the dynamics is chaotic in some sense, or supposedly at least. So the question, one of the problems was to try to understand better this kind of picture in the general setting. Because locally, when you have this kind of phenomenon, the elliptic islands, they are very well understood from the local point of view. It's KAM theory. But for instance, if you take bigger and bigger invariant circles here and you take the biggest one, the maximum invariant circle containing this point here, the topology of the boundary of this circle, well, we don't know much about this, the dynamics as well.  But here, in the maximal invariant disk, we don't know very well what can happen. We don't even know a priori if this disk can be very large and bounded in a way. So this was one of the motivations. Well, and there are other related questions. For instance, when you take the stable manifold of a hyperbolic periodic point and you look at its closure, you have a compact connected set which is invariant. And then the connected components of its complement, well, they are open. If the maps are preserving, they are open periodic sets. And these sets have a nice boundary in the sense that I was saying there. So you can try to, if you want to understand the closure of the stable manifold of a periodic point, it makes sense to study the boundary of the complementary invariant sets. So that was also one of them. Okay, so let's consider the model case which is when you have this simply connected invariant set and its boundary is a circle. So it's a disk, my set. It's a closed disk. So in that case, everything is nice. We have dynamics, the boundary is a circle. So the dynamics in the boundary, well, you can define the rotation number and you can say as everybody knows that if the rotation number is irrational, then you have no periodic points in the boundary and you have a semi-conjugation to a rigid rotation on the circle and you have a unique minimal set, which is either the whole boundary or a counter set. In the case of the rotation number, it's a rational p over q. And then you know that, well, there are fixed points for fq. In fact, all periodic points are fixed points of fq. Moreover, all non-wandery points in the boundary are periodic points of fq, right? And in particular, this implies that when you take any point in the boundary, if you iterate it, it converges towards a periodic orbit of pq. In the future and in the past. So the dynamics in the boundary is very, very simple in the rational case and the irrational case is basically very similar to the rotation. So the question is, can we reproduce this for a general boundary when it's not a circle? And here, the point is that the boundary can be really complicated whatever I was saying at the beginning. So here is just a simple example. This is not so complicated, but it's already not a circle. This is not locally connected. But then you can have something like this, which, okay, it's still not so complicated, but a bit more, or something like this. So the open set here is the region, the complement of this spiral, right? So now all the points in this outer circle, they are very bad because you can't go towards them with an arc from the open set. But then you can have... The outer circle is part of the boundary of you, yes. It's just an example anyway. But here is a much better example, which is called a canister continuum. I think it's nowhere locally connected. And this can be the boundary of a disc. I mean, in this case, the disc would be the disc containing infinity in the plane. So it's the boundary of this disc. And it's very far from being a circle now. Now this really, I can't... Because this one, well, you can say, okay, but there's some circle there. But here, it's really nothing to do with a circle. And then you can have things like this. This is the water legs continuum. The continuum is the black part. The blue parts are one, two, three. Three invariant discs. Not invariant, there is no dynamics. There are three discs, topological discs, which they share a common boundary, which is the black part. So the boundary of any one of these three discs is the same set. And this set is, again, nowhere locally connected. It's really bad. So if you look at the complement of this set, there are three discs plus the unbounded disc. This set, in particular, appears very often just like this one appears. This one appears or something very similar when you have a homo-clinic intersection. You take the unstable manifold and you look where it accumulates, you get something like this. Here, this appears when you have something like a plucking attractor, a hyperbolic attractor. You always get pictures like this. If you are on the torus, you can have something like this where the open set is the black thing, the dark thing, and you have a virus accumulating on something which locally looks like a counter-set times an interval, also very far from being a circle. So that's the point. I mean, the point is you can't just say, OK, I will define a rotation number and reproduce what we do in the circle because the boundary can be very different. Here's one of the worst things that may happen in a way, which is the pseudo-circle, which is a compact connected set, which is the boundary of a disk, but it's not only nowhere locally connected, it doesn't contain any arcs. So whenever you have a continuous function from the interval to this set, it has to be constant. So it's really bad from the topological side locally, at least. When you zoom in, you always see things like this. You zoom in more. You see things like this. There's nowhere. You can't find an interval in there. OK. So what do we do? How do we define? Sorry? All of this. So this one, the one of the legs, it's a plucking attractor. So it appears as a hyperbolic attractor. This one also appears dynamically and this one can also be realized. There is a construction by Handel and also an example of Michel Hermann, where they do it like an attractor, or even as an hyper-serving environment set. Which, this thing? This one. The basin of a... Yeah, why not? Oh, yeah. You mean instead of three? Yeah. As many as you want. OK. The interesting thing about this kind of object is that they appear robustly because they appear as hyperbolic attractors, for instance. So they are not like pathologies. They are something that you can't avoid. And in any degree of smoothness. Sorry? This one? Yeah. It's an open disc. Yeah, it's an open disc. No, no, but this is the boundary of the disc. In the boundary of the disc, you cannot find any arc contained there. Any image of an interval. OK, so how do we define a rotation number? So the trick is to use a compactification which replaces the boundary. So you take the open disc and the boundary may be very bad, but you can just replace it by an artificial boundary which is nice, which is a circle. And if you do it right, the dynamics extends to this compactification and then you have a circle of a morphism and you can define a rotation number. And then you can try to reproduce the Poincaré theory on the circle to this new setting. So the way of doing this is using Caratt d'Odori's primance compactification. And it was done the first time I've seen it done dynamically. It's by Cartwright and Litterwood in the 50s. And so you can, if you do this, I will explain it here. So what you do is, so the idea is instead of looking at the boundary, you look at the directions. How can you approach the boundary from within the open set? Look at directions. You order these directions and you get something like a circle. And you put a topology there, that's roughly the idea. So there are two ways of defining this formally. There is a dirty way which is just using the uniformization of the disc. You just take the disc which can be very complicated and you have the nice disc. And you have a map here which is a Riemann map. So you have a dynamics here. And if you conjugate by this map, you get a dynamics here. And this dynamics, you can show that always, as long as this map is defined in the closure of you in this bad boundary, you can always extend this map to the circle here using a conjugation. So it's just a change of coordinates. So this gives you the nice boundary. Just forget about whatever is outside of you. And you look at it on this coordinate and you look at the closure. That's it. And the map extends. Here, yes. Yes, the primates will end up corresponding to these points here. But this is absolutely artificial. I am defining primates in this way. But there is a more topological way of defining the primates which is usually more useful. Just a remark is that of course this map doesn't extend to the closure here because this boundary here is maybe not locally connected and this is locally connected. But the point is that the dynamics does extend to the closure. So the topological way of defining primates, which is the more useful one, is to do it, I will choose a marked point here just to simplify. I mean, we don't usually do this, but it's easier if I do this. I just fix a point here. And I will define some things. So a cross cut of U is just an arc joining two points of the boundary of U which is otherwise completely inside of U. Just the end points are in the boundary and the arc is inside of U. So this is a cross cut. And then a cross section is one of the halves. This thing divides U into two halves. So I will denote by D of alpha the cross section which is the one not containing P. That's why I chose P just to have a reference. So this is the cross section D of alpha. And then you can order these cross sections. If I have another cross cut here and I have D, this is alpha prime, D of alpha prime, then you say that alpha prime is smaller than alpha because D of alpha prime is contained in D of alpha. So this gives you an order between cross cuts by looking at the corresponding cross sections. A partial order. So the nice thing is that of course cross cuts are mapped to cross cuts by the dynamics. Everything is nice. And so to define a prime main what we do is we look at nested sequences of cross cuts or cross sections. So you just take a cross cut here, a smaller one here, a smaller one here and so on. And you look at the corresponding cross sections. And you require that the diameter of the corresponding cross cuts goes to zero. And you look at this chain of cross cuts, modulus and relation because you will have many ways of representing the same object. I will show a picture soon. But the relation is the following. I will say that two chains of cross cuts are equivalent if they have the property that whenever I fix one element of one of the chains, the elements of the second chain are eventually inside of this one and vice versa. They go towards the same direction in some sense. So in a picture here, so this is a chain and this is an equivalent chain. They define the same thing. But of course this is a part where the boundary is nice. Here is a part where the boundary is not so nice. But here is a chain and this defines a prime. The diameter goes to zero, so it defines a prime end. But you can see here that prime ends don't correspond to points of the boundary necessarily because here what would be the point of the boundary corresponding to this chain. You have a whole segment here that you are accumulating in. And there are more complicated things but I can't show nice pictures of them. So this we have to do. So the point is you define the set of prime ends as the set of all chains, decreasing chains of cross cuts, modulo these equivalence relations. And you can show that this is a circle if you put the right topology there. So that's the topological way of finding and it's equivalent to this one. Okay. But what do you mean? What are the prime ends? It's a circle. I can't show you a prime end. So you just draw a sequence of things with diameter going to zero. No it's really, I mean there are no nice pictures that I can show you too. Maybe, maybe, okay let's, you mean here. Well here I can show you, here you have some channels. You see for instance the darker circle it has like a tongue that goes and spirals and accumulates everywhere. So this tongue it will correspond to one prime end in a way because you can choose a sequence of increasingly smaller cross cuts which goes away towards the tongue and then this defines one prime end. But this is an easy way. Here it's harder to find an image of what a prime end is. Okay. No, no, no it's just a whole meal. Yeah, I mean even if the dynamics here is smooth everything is zero here. But even, yes, yes, it's a whole meal. Yeah, in fact there's a nice thing that here if the dynamics is holomorphic then here it's holomorphic as well. That's a nice thing about the, but the diameter of alpha is that goes to zero. If you look at the cross section it can be really huge. It can be that for all the cross cuts, the corresponding cross sections are huge and you can't avoid it. The intersection of the cross sections. Well it will be empty but if you take the closures and you intersect it could be that you always get the whole boundary for instance. This is what happens in the pseudo-circle. So it's really, it's really bad. Yeah. So the problem is now that we defined the prime ends. So you have a dynamics on the prime ends because f maps sequences of cross cuts to sequences of cross cuts everything is nice. So you can do the dynamics in the prime ends. You can define a rotation number there. And the question is now how do we relate this rotation number with the dynamics in the boundary? The real boundary, not in the prime ends. So for instance the first basic question that you can ask is if the rotation number in the prime ends is zero then do you get a fixed point in the boundary? And well in general the answer is no. Here's a picture. So the disk is something that spirals towards another disk here in the future and in the past and the dynamics flows from one to the other. That's all. It's just like a translation but this is a rotating disk. This is a rotating disk and here the dynamics flows from one to the other. My disk here is this smiley thing. So what happens is the boundary of view doesn't contain any fixed points because this boundary side flows towards there, this flows towards there. Here everything rotates, here everything rotates. In the boundary of view there are no fixed points. But if you look at the prime ends, the dynamics in the prime ends is like this. The circle of prime ends is the dynamics. It's a north pole, south pole. And you can clearly see this because if you take a cross cut here its image goes inwards. So this must happen also in the circle of prime ends. So if you take a cross cut here it goes inwards. So everything flows from one side to the other. So in the prime ends you have fixed points. So the rotation number is zero. The prime end rotation number is zero. But in the boundary you don't have fixed points. So this is bad. But the observation here is that you have this situation where you have a cross cut which is mapped strictly inside, inwards. So it's kind of like an attracting region. So I will call this a trapping cross cut. When you have a cross cut which has the property that its image goes inwards. Either to the future or to the past. So what happens here is that you don't have a fixed point but you have a trapping cross cut. And in fact, Carter and Glitterwood in that paper I mentioned before they showed that it's actually a very simple argument that if you have a zero rotation number in the prime ends then either you have a fixed point in the boundary or you have a trapping cross cut. So the only way it can fail is that you have this situation where something is attracting things towards the boundary. And in particular, if you know that f is area preserving for instance you can't have this. You can't have trapping cross cuts if it's area preserving. So in the area preserving setting, things are nice. Zero prime end rotation number implies fixed point. Okay, so... Yeah? Sorry. Okay, so in your example... Yeah? Yes. So zero implies in the boundary there is a very unregional reason there is not a full boundary. So yeah, so the... One thing that we did with Rata here is we showed that precisely this is the only bad situation that you can have. If you have zero rotation number and you don't have a fixed point in the boundary then you must have some kind of sets which are attractors and repellers and they are rotating in a way and the boundary of you is contained in the basings of these things. So it's basically like this. Like both dynamically and topologically is very restrictive. So it's really an exception in this kind of situation. But okay. Was that the question? Yeah. Okay. So what about the converse? The point here, the point to remember is that in the area preserving setting things work nicely. So the converse... By the way, I don't know if I was clear enough at the beginning but I'm always assuming that you is on the plane and is a bounded invariant set by a home of the plane. I'm just simplifying. I don't know if I actually mentioned this but it was written there. So the converse is if the rotation number is not zero can we say that there are no fixed points? This is the other natural question because this relates to the question of whether when the rotation number is irrational you have no periodic points for instance. So this is much more difficult and in fact in general the answer is no but it's trickier to make examples. You have this example by Walker which is like you have some sort of Denjois example here with hairs spiraling towards an outer circle and in the outer circle you have everything is fixed point-wise. So these hairs are rotating with the combinatorics of an irrational rotation but the outer circle where they accumulate is all fixed points. When you look at the primates of these you will see the Denjois dynamics. So you have an irrational primate rotation number but you have a bunch of fixed points which are in the boundary of U. U is of course the white part here. So this is Val. They are not accessible, right? You can't reach them by means of an arc containing U. Yeah. In fact you can't... Yeah, I'm not sure if you can make an example where I don't think you can make an example with accessible fixed points. You can't. You can't. But there are other examples which are in fact in some sense more interesting. There's a bulk of attractor which is something where in the annulus you have an attracting set which contains like the annulus goes inwards and you have points there which rotate with different velocities in the annulus like the rotation along the... So this one for instance a fixed point and a fixed point which rotates one. And then when you look at the attractor you get something which is really topologically complicated. It's nowhere locally connected and so on. But the dynamics contains all sorts of periodic points of all different periods and different velocities and so on. So it's a very... And this appears as the boundary of a... It's a basing of attractor in a sense. So in this kind of example you get... You can get non-zero rotation number and fixed points and so on. The point is that all these examples they have the property that they are attracting. Like this example you can't... If you try to build it you end up getting... It's going to be an attractor basically. The boundary is going to be attracting. So again if you try to make it a preserving you can't. And the first... The first work we did with... Patrice and Maison was to prove that in fact this is the case that you can't do it unless you have some sort. It's a similar thing to the result I mentioned before. Either you have... You realize the zero rotation number of fixed points or you have a trapping cross cut. Well here it's similar. If you have a rotation number then either you have no fixed points in the boundary or you have boundary traps. I will define them just in a second. But it's something similar to what I said before. And in particular it's something that can't happen if the map is area preserving. So as a corollary if the map is area preserving when the rotation number is zero or more generally when it's rational you have a fixed point or a periodic point and when it's not you don't. So it's really... It's exactly as in the circle this time. So again the area preserving setting is... Everything is nice. So what is a boundary trap? So it's similar to the notion of trapping cross cut but of course you can't have a cross cut that is mapped inwards if the rotation number is not zero. Because of course if you have something here that is mapped inwards then you will have a fixed point in the boundary circle. But then... Boundary trap is just a union of cross cuts such that the region they bound is mapped strictly inwards either to the future or to the past. Basically it's that with some technicalities there. That's... In particular of course this is not compatible with being area preserving because it means that you are attracting something towards the boundary. You can't be non-wandering. This thing could be wandering, yes. Or not. But the point is it implies that arbitrarily close to the boundary there are wandering open sets for instance. So it's very... It's kind of restrictive in a way. So we say that the map satisfies the boundary condition in you if there are no boundary traps for any power of the map. In particular if f is area preserving it has the boundary condition. Or if it's non-wandering or something like this it always has the boundary condition. And another case... The nice thing about this is that the boundary condition also holds even if you don't assume any kind of area preservation or anything like this but if you assume that the map is for instance holomorphic as I was mentioning before and the rotation number is irrational then you automatically have this condition by completely different reasons. So it's not necessarily a condition that is guaranteed by recurrence or area preservation it could come from something. So the point is putting things together is that if you assume the boundary condition then the... Sorry? If you have a holomorphic map with an invariant disk it has to be invertible Yes? Yeah, but it was happening and we have a single disk So putting things together you have the nice picture for the area preserving setting for instance. You have that if the rotation number is rational you have a periodic point only of period Q exactly as in the previous case and if it's irrational then you have no periodic points. But of course in the circle we have a lot of extra information we know in the irrational case you have a semi-conjugation in the rational case you have this thing about all orbits converging to the dynamics being very simple and so you can ask whether the same things hold assuming this boundary condition or area preservation and in general you don't in the irrational case you can't expect to have a semi-conjugation to an irrational rotation for instance the pseudo the pseudo-circle the examples with the pseudo-circle with the irrational rotation number the hundreds example in particular it's not semi-conjugated to an irrational rotation but but in some cases you can with some topological condition which is a bit strong but not so strong you can you can get the semi-conjugation so some partial result in this direction was done by Tobias Jagger and myself but I want to get into this then you can ask whether when the rotation number is rational you know that the dynamics in the boundary is trivial in the sense that everything flows from periodic points to periodic points no no even in the holomorphic case you don't have that you don't have a semi-conjugation in general you don't if you do you mean are you talking about this thing this has nothing to do with being holomorphic it's a topological condition in the boundary semi-conjugation no even if it's holomorphic no in general no if you know something about the topology of the boundary you can sometimes show the semi-conjugation but even in the holomorphic case you don't have a semi-conjugation so the second question is whether when the rotation number is rational you have these trivial dynamics and this is nicer because the answer is yes and this is the second part of the work with my salmon patrice is basically showing that in the rational case everything behaves exactly as in the circle there's no exception if we assume the boundary condition it's a fixed point so let me state a theorem which is basically the important part that implies everything is that if the rotation number is zero assuming the boundary condition then every no wondering point of the map restricted to the boundary is a fixed point so if you know this in particular you know that every point of the boundary from fixed points to fixed points because the omega limit of an orbit is contained when there is set so it has to be all fixed point so the boundary dynamics is trivial in fact so that's the main theorem basically but actually the theorem is a the proof of the theorem goes through some topological result which is actually the sense of the proof which is a bit tricky to state but I will try to explain the the basically it says that so you have this open set on the plane it's bounded but it's complicated I don't know and then you have an unbounded component here that I called U infinity U infinity is the unbounded component of the complement and you have U and then what the lemma is saying is that if the rotation number in U is 0 and the map is has the boundary condition for instance if it's reperserving then what happens is that the boundary of U infinity is either all fixed points just the identity basically or U is compactly generated so compactly generated means that if you remove a point from U here and you look at the point of infinity you are on an annulus you can go to the you can leave this to the universal cover and look at the boundary of U in the universal cover of this annulus and you will see something like this I don't know this is the lift of U and this is the lift of the boundary and in general the lift of the boundary can be disconnected even though it's connected here when you lift it it can be disconnected for instance in the spiral example this example like this when you remove a point here and lift it to universal cover you will see a line which is this circle here and a bunch of lines accumulating like this this is the lifted boundary of this thing so it's not connected it has parts which go a bunch of lines so being compactly generated means that this doesn't happen it means that you can find here on the lift a compact set in the boundary that when you project it you get the whole boundary here you can't sorry a compact connected set here if you take a compact set that projects to the whole boundary it's not going to be connected so this is technical but it's really a strong thing to say that the boundary is compactly generated and it's actually the sense of why the other result holds because basically what you do is you use you take a point which is no wandering in the boundary and you use some recurrence properties to show that the boundary has to kind of twist and spiral a lot then it can't be compactly generated it's like this so basically the topological part is the important one okay and for instance a corollary of this topological result which I find really nice is that if you take another preserving map which leaves invariant a pseudo-circle like the Handel example if you have a fixed point in this pseudo-circle then the whole pseudo-circle is the identity the dynamics of the pseudo-circle is identity but to me that is surprising because in fact one thing is important is that it's not like the topology of the pseudo-circle implies that the dynamics has to be trivial because the Handel's example has a non-trivial dynamics it has an irrational rotation number and it's not semi-conjugate to a rotation so it's kind of complicated in a way but if you have a fixed, something is fixed then the dynamics has to be the identity so just to mention another consequence of this last result of the dynamical result one of the one of the motivations for our first work was previous work of Mater where he showed that if you take a diffeomorphism on a surface a CR-generic diffeomorphism which is that of preserving the prime and rotation number that disk is always irrational this is not a dynamical result because it's talking about the prime and rotation number there is no relationship a priori between the prime and rotation number and the dynamics of the map so it was kind of an abstract result in a way but then of course putting it together with the results I mentioned you get the dynamical information which will tell you for instance that in the boundary of this disk there are no periodic points if you take diffeo if you take an invariant disk and you look at the boundary there are no periodic points and the proof of this which is really useful for some things it relies on the following generic conditions you need to assume that the fixed points are all hyperbolic or elliptic and that there are no saddle connections these two are kind of standard conditions that you can require but then there is another condition which is that the elliptic periodic points have to be generic in a way which is either they are surrounded by invariant circles with the rational rotation number or they are surrounded by like homoclinic intersections of a periodic point which go around the point any of these two conditions works but both of the conditions are much harder to guarantee than the other ones in a way so the nice thing about the second result the one that I just mentioned is that you can get a more precise version of these which basically doesn't require any generic conditions you can explicitly state the conditions take another preserving map at least in the neighborhood of the boundary of an invariant disc so I have an invariant disc by another preserving map which is C1 and suppose you have a fixed point here this fixed point has to have real positive eigenvalues in particular it can't be elliptic you can't have elliptic points in the boundary of a disc and this maybe it sounds obvious because elliptic points rotate or something but it's not really obvious and the point is I am not assuming any generic condition the main reason to assume this condition here or this condition these things was to remove the elliptic points from the boundary but the point is you don't need any condition they can't be in the boundary this is not a generic thing and the proof is really easy I will I will explain why this is true so if you have if you have a point here of course I want to prove that there is a positive eigenvalue if there is a positive eigenvalue it can't be elliptic because by the way it could be that the eigenvalue is 1 this could happen but it can't be models 1 different from 1 so suppose this has non-real eigenvalues you can blow it up you can blow up the point to a disk and you will have some dynamics here which is defined by the derivative of the map and the open set here will accumulate on this circle in some way it depends on how the disk was accumulating in the point but it will accumulate at least at some points of this circle and so when you look at the boundary of view intersected with this new circle blown up you will get something which is an invariant set in this circle in particular you will get a recurrent point there there is a recurrent point in this circle right but the rotation number in U is 0 I'm assuming that sorry I'm not assuming that but okay I don't need to assume it because I had a fixed point here so the previous theorems tell me if you have a fixed point the rotation number must be 0 so the return number is 0 so the thing is you have a recurrent point here and there was this previous theorem which says every no wandering point in the boundary is fixed so what happens is that any recurrent point here in this circle which is in the boundary of view must be fixed and there is at least one so there is a fixed point there but this was the blown up dynamics so the dynamics in this circle is the projectivized the normalized dynamics of the derivative so you have a fixed point there so this means that there is an eigenvalue an eigenspace so you have a real eigenvalue and so so that's the proof it's really straightforward so as a consequence if you assume that I mean because remember that's why we have more general hypotheses you don't need area preserving you just need the boundary condition and because the map was area preserving here even if you blow up and you change things the inside of you you didn't change much it still preserves a measure which is positive on open sets in you so it has the boundary condition boundary yeah that is the reason that we do these more technical statements because when you try to make surgeries and modify things you lose the area of preservation sometimes or you lose the general condition but in technical conditions you can usually keep so as a consequence if you assume that the fixed points are all hyperbolic or elliptic when you take an invariant disk you either have no periodic points or if you do you can tell not only that the dynamics in the boundary is simple in the sense that everything goes from from periodic points to periodic points but also it's actually a union of saddle connections so it's really like you have here a saddle point wait saddle point whatever and so on a bunch of saddle connections maybe you have something like this also I don't know so this is the boundary of you but in particular in the rational case it has to be locally connected because it's an arc the union of saddle connections it's an arc and this is not a typical it's not very common to have this so it's telling you that if the rotation number is rational not only the dynamics is trivial but the topology is really very simple under this assumption that periodic points are hyperbolic or elliptic well it's fixed points here how much time do I have how much oh by the way this result this last result here was announced also by Fernando Oliveira 10 years 12 years ago but still a published but and the proof is completely different the proof he had so putting everything together we have this nice picture for the area preserving setting which is similar to the picture I showed for the circle even if the boundary of my disc is complicated I can tell that if the rotation number is irrational then you have no periodic points and here's a thing that I didn't mention but you can say something about the topology of the boundary you can say that it's annular which means it's a nested intersection of topological annular so it's some kind of topological condition and if the rotation number is rational it's the nicest situation because it's exactly like in the circle so you have fixed points in the boundary for fq every no wandering point goes from it's fixed and in particular every orbit goes from fixed points to fixed points and you have the additional thing that I mentioned that the boundary is compactly generated in the sense that I explained here unfortunately no basins are the problem because you see the counter examples I mentioned at the beginning where all the problem was that they were basins they were attracting or repelling so yeah you can make like the brick of a tractor it's a counter example to everything I'm saying and yeah and it's amazing so you can't improve this in this sense so I wanted to mention I will skip this but so the proof of these results they rely on one of the main things that we use is for the latest result is a result of Bégan-Crovisière and Leroux about the existence of maximal isotopes which is related to previous work of Patrice Le Calvé and we use this to precisely to relate what I said before that when you have a point in the boundary there is no wandering in the boundary this implies that the boundary has to be topologically complicated it has to spiral in a way to obtain this it's really tricky and we need to first choose specifically some maximal isotopes and then do some define some rotation numbers associated to the isotopes and work the way towards the topological results using some linking lemmas and so on so I will try to show I'm not going to say anything about this but I will try to show one of the ingredients of the proofs which I like that has to do with the shadows of arcs the idea is you can define the shadow if you have an arc if I have a my disk here and I have an arc let's say a loop I can look at the cross cuts defined by this arc here is one here you have some other one here alpha 3, alpha 4 when you look at the prime ends compactification of view these cross cuts will show up somehow here alpha 1 alpha 2 alpha 3 like this we have some fixed point here reference point so we define the shadow of this arc gamma the shadow of gamma is going to be a subset of the prime end circle which corresponds to the prime ends which are obstructed by gamma from P so is this interval together with this interval it's like if you put a light source here the prime ends that become dark are the shadow not exactly but the light bends yes so the basic lemma just 2 minutes is that if you take an arc like this which is this joint from its image by F so it's free by F then what happens is that the the shadow of the cross cut generated by gamma is contained in 2 intervals where the dynamics is just a translation this is really restrictive this is if we assume the boundary condition or if we assume that gamma doesn't contain any boundary traps so what this is telling you if you look at the dynamics of the prime ends for instance if the rotation number is 0 you have maybe you have some fixed points here and between the fixed points you have some dynamics like this which is translations and then what this is saying is that the shadow of any free arc which doesn't contain a boundary trap must be contained in the union of 2 intervals of this kind and they have to be intervals with different directions so this is maybe this is one and this is one so all the cross cuts must be here you can't have one here or here and this is really restrictive and this is the key to proving the topological results because when you have this there is another result which is the maximal cross cut lemma which tells you you can find certain cross cuts here which are maximally in a nice way and then what you do is you find one maximal cross cut here one maximal cross cut here and you get its image which flows this way and its image sorry I'm trying to explain the proof of this actually so I want I want because I'm out of time but the point is when you locate the shadow of their gamma you can then use this to prove something similar on the lift to universal cover and the compactly generated property what I mentioned before and with this and the linking lemmas and the maximal isotopes you get the the dynamical sorry I'm out of time so I will stop here you'll find an explication of fundamental attract for example plate and then its cross section on the Lovacevsky plane the connection the connection between the absorptors I'm not sure I don't think I've seen this for the plucking attractor I'm not really sure you can see the dowel of this the absorptors and then it's possible to find the rotation number on this absorptors some similar yes it's probably just a way of interpreting there are many ways you can do this so probably it's the same thing yes oh so it's not invertible okay she knows I don't know you have to ask Juliana or Leyva because I've never tried I mean but I know they have something yeah it's very difficult I'll be interested in realizing what's happening for a man you may happen that the impression of the big brain is the whole set and it's not that familiar but you don't know how to find the cross section okay that's a good question


Joining us today on the web show are Buddy and Meg. This is Meg. Buddy is on my lap. He'll probably be licking us during the whole show. He sounds like he's going to peek up any minute, to be honest. So excuse us. Is that a good thing? They just don't want to be alone. Did you catch last week's episode? It's always lovely to see what goes on behind the scenes. Who made those incredible voices, the incredible puppeteering?  Anyone who is a short circuit fan, whether it's a black fan, a slightly navigated fan, make sure you check out last week's episode. There's a card above. I'm trying to wrangle Buddy at the same time. We thought we would do another giveaway. People like getting given stuff. Getting given stuff. Getting given stuff. Sorry, that was hard to say. I know we've already done a Flash Gordon competition before, but this is, that was the Arrow North American release. We are giving away the five disc UK release from Studio Canal. So this is different to the one we gave away a couple of weeks ago. And if you were wondering how much Flash Gordon can we literally cram into, or how much Flash Gordon could you actually take, well, there's more. Five discs of it literally crammed in your greedy throats. So also on this disc, there's like a mini booklet of the Flash Gordon book, a little sample coming out, the Queen soundtrack, Life After Flash is on it. Life After Flash is on it. There's a patch. There's art cards, a bunch of extras, different to the other one. So we are giving this away and... What else are we giving away? Just some more little bonuses we're going to throw.  So Arrow are really on this kind of 4K Blu-ray. I don't know if they're all 4K, but there's a bunch that we're going to throw into. American Werewolf in London, Incident in Ghostland, Last House on the Left, Kansas City, ICU. That's got to be over £100 worth of giveaways. It may be £200. And we could do some kind of crystal set for... What are they used to do on Bullseye? Oh, the... what do they do? A speedboat. A speedboat! Do you want to keep these or do you want to go for the speedboat? Oh, I don't know. M1. There'll be no more washday blues with this combined washing and drying machine. M2. Your score 11's is out of 10 with this espresso coffee maker. M3. You clean up with this. It's a high pressure cleaning system. What did sort of, you know, Terry and Norma from Birmingham want with a speedboat? I know. That's her parents' names, you know. She's an example. You won't understand bulls-eye, but if you haven't seen it, you should watch it on YouTube. It's brilliant. So we're giving away all of these. Pretty impressive, I might add. So what do you have to do to win? First of all, you have to subscribe. This is a competition only available to our subscribers, and we will check. So get the subscriber button clicked, the notification bell. If you go back and look at all our rather 13 videos that we've put up, you'll see the kind of stuff we're going to keep. I might have seen Brian May a couple of weeks ago just hanging out with us for half an hour, telling us what he's been up to. No, a half a way. Yeah, Sam Jones, just hanging out. Joey Prenner. All of our celebrity buddies. So please subscribe anyway, because we've got so many more great things coming, but subscribe and comment to win. What do you have to comment? I don't know, I've forgotten. What do you have to comment? He had won two jobs that hold... Well, one to stop Buddy puking up any second, which I'm managing at the moment, so that's one job. The second one was to hold up the blue rays. What was the third one? Comment below. Oh yes. The 4K thing. What film would you like to see re-released in 4K people? Seems to be a trend happening in the moment, a trend that we're loving that all these 4K re-releases are coming out. What would you like to see and why on 4K? Why's the important bit?  We're going to actually be picking the answer, not just randomly from a habit. We're going to be choosing the best one. So make it good, make it count. If you comment more than once, you've got more chances of winning. Hey, and it helps the algorithm. And I hope so. Thanks, YouTube, for having a really complicated algorithm. We're trying to navigate every week. There we go. So that is it for this episode. We are back next week with another interview. We wanted to just mix it up a little bit, because it's also the last weekend of August. It's a bank holiday weekend, people. It's a bank holiday. So I hope everyone has really great plans for the bank holiday. This goes out Sunday, so you still have one more day. Enjoy. We'll see you there. Until. Until next week, when we'll be back for more. Have a great week and we'll see you soon.


According to statistics, only a small percentage of you who watch our videos are actually subscribed. If you're not subscribed yet and you enjoy what you see, do consider hitting the subscribe button. This encourages YouTube's algorithm in promoting more of our mental health content to more people out there. Hey Psych2Goers, and welcome back to our channel. Why do most bad situations you face take a turn for the worse? What do you do when you're overcome by a powerful emotion? Do you still end up acting out on them despite knowing better? Especially in the heat of the moment, it's easy to get carried away and end up doing or saying things that you later regret. Yikes! And with whom does the blame lie? Emotions. Your emotions are vital. They tell you a lot about your surroundings and how to best move forward. But how to tell when they begin to control you all together? Here are five signs that you might be acting out of emotion and that it might be beneficial for you to take a step back and process your emotions prior to moving forward. One, you focus on getting back at them. Are you often trying to create in them the same feeling that they instilled in you back then? The saying, two wrongs don't make a right, really holds true here. In a burst of emotion, you sort of retaliate. As a result, you fail to realize that this kind of response really doesn't help the situation at all. It continues to foster ill will and does not allow for the situation to cool or a solution to present itself. Two, you hear rather than listen. Are you in the habit of allowing others to speak just so you can deliver a response? When you hear them out just so you can formulate an answer, you're not really listening. You are unable to truly grasp and understand where the other person's coming from. You may be using that information that they give you as ammo to strengthen your points and shoot it right back at them. It might be something to keep an eye out for as listening without understanding causes one to jump to conclusions. You may feel to see the bigger picture and end up acting rashly. Three, you play the blame game. Are you quick to believe that everyone else has wronged you and that you've done everything perfectly? When emotions run high, you become insistent upon proving the other wrong. This can cloud your judgment of who's its fault. It makes it difficult for you to introspect and own up to your mistakes. As a result, you grasp another shoulder to shift the burden on to, thinking, maybe I did something wrong here can help prevent things from blowing out of proportion. Four, you're set on mulling over the details.  Do you find it hard to forgive and forget? If the solution seems out of focus, that's because you're more focused on what happened and how it made you feel. And this isn't to say that you're not allowed to express your emotions. You definitely are. But if you're constantly focusing on the negative and not striving to come to a resolution or a better way to move forward, that might be a sign that you're running away with your emotions at that point. Five, there is no pause between intense emotion and action. So think of the last time you experienced a really intense emotion. Did you act on it immediately? Chances are you did not pause to ponder. Oftentimes, when you're acting from a very emotional mind, it lacks logic. Even though emotions tell you how to best respond to a situation, if it's a really intense one, sometimes that reaction or emotional response will elicit something in you even you yourself don't recognize. And so you act in a way that is so not you in retrospect. As the saying goes, think before you act. This shows taking a step back to really think about what just happened and how you should respond helps ground you in the present rather than letting emotions fly high. So should suppressing your emotions, be it out of fear, out of embarrassment, or out of shame, be your preferred response? Rather, it is prudent to best utilize your emotions in an effective way. By learning to recognize, process, and express these feelings, you can stay true to yourself. Even in the face of extreme emotion, staying connected with rational thought helps you reach a state where you're a bit calmer and can respond with a more level head.  Do any of these describe your experience? Or are there any signs that aren't included in this list? Leave a comment down below and please feel free to share any thoughts you have as well. If you find this video helpful, be sure to hit the like button and share it with those out there wondering why they always see red. Don't forget to subscribe to Psych2Go and hit the notification bell for more new videos. Thanks for watching!


Hello from the National Archives Public Programs and Education Team. My name is Missy McNat and I'm an Education Specialist in Washington, D.C. And welcome to the National Archives Comes Alive Young Learners Program. You can find information about future programs on the National Archives website, archives.gov, and on the National Archives Facebook page. This morning we meet Madame C.J. Walker, portrayed by Dr. Daisy Sentry, author, educator, and actor with the American Historical Theater. Madame C.J. Walker epitomizes the American dream. She was born Sarah Breedlove in 1867 in Delta, Louisiana, and she was the first child born free to her formerly enslaved parents. And through her dedication, hard work, and perseverance, Madame Walker built a hair and cosmetic manufacturing business. And in the process, she became the first African American female millionaire. And that was no easy task in the Jim Crow segregated and male dominated society of the late 19th and early 20th centuries. But Madame C.J. Walker was so much more than an entrepreneur, and we will learn about the many roles that she had during her life this morning. The National Archives has in its holdings records related to Madame C.J. Walker, including these two photographs that you can find in Doc's Teach on her estate, Madame C.J. Walker's estate, Villa Loireau, that is located on the Hudson River in New York. And on the next slide, we see the feature activity for the program today on Madame C.J. Walker that you can find in docsteach.docsteach.org. And we will share this slide again at the end of the program. We are very fortunate this morning to have Miss Alelia Bundles with us, who is Madame C.J. Walker's great, great granddaughter. And Miss Bundles, among many other things, has written several books on her great, great grandmother. And this one, all about Madame C.J. Walker, is a book for young adults. And you can find this book. You can purchase a signed copy from the National Archives Foundation gift shop by going to NationalArchiveStore.org, go to books, and then go to signed books to get your signed copy. And Miss Bundles is going to answer questions for us this morning at the end of Dr. Century's presentation. So please write your questions in the YouTube chat box for Miss Bundles to answer. And we have a National Archives staff member who is monitoring the chat box. And let us know where you're watching from. This program is brought to you by the National Archives Public Programs and Education staff, the National Archives and the National Archives Foundation. And now it is my great pleasure to introduce to you Madame C.J. Walker, a remarkable woman who changed the course of her life and thousands of others who were associated with her. Enjoy. I wasn't born with a silver spoon in my mouth, no spoon at all, I guess. Mama said I was going to be her Christmas baby, but of course I couldn't wait for Christmas. I came December 23rd, 1867. Mama said I'm going to be lucky because I was born free. I was given the name of Sarah, Sarah Breed Love. As far as I can remember, me and my sister, Lavinia, you were playing the mud because it rained a lot down in Delta, Louisiana, down on the Barnas Place. As young as I was, Mama said, Sarah, get that book learning, Sarah, learn how to read, write, and count, and mind nobody else's business except your own, you hear? I remember that. Then all of a sudden my mama died, and daddy died right after my mama. When I am about seven, I guess, often. Me and Lavinia, we tried to fend for ourselves the best we could. Then we took the paddle boat across the Sampy River into Vicksburg looking for my brother. But we ended up, most of the times, in the Washington woman's shanty shack. Wash and close, sunrise, sunset, hands all wrinkled, backs on fire, feet on fire. Then I got married out of convenience, I guess, to get away from my mean brother-in-law. I got married to a nice gentleman by the name of Mr. Moses McWilliams. He worked at the Sawmill on the other side of town, but one day Mr. McWilliams went to work. He never did come back. We suspect he was limped. We ain't sure, but back then he wasn't safe for a colored man to be out. I never did talk about that for the rest of my life.  I named my little girl Leah. I said, I'm going to make sure that little girl learned to weave, write, and camp. So I got on the paddle boat, got off for the first stop, sent room. Didn't know nobody, didn't have no job, no money, nothing. But I went to the church and they had rooms to rent. And she said, whenever you get the money, you can pay me. Oh, that was a godsend because I didn't have no money. And the church had a school. So I would put Leela in school while I would wash all day long, but I learned to read, write, and camp right alongside Leela.  Oh, I continued to wash clothes and saving every penny. And a gentleman caught my eyes. His name was Mr. John Davis. And after a while, Mr. John Davis and I got down. He was bringing in 250 a week. I was bringing in 250. I said, John, how are we going to afford a big house like that one up on the hill? And how I'm going to order nice dresses from the Cadillac. And how are we going to get furniture? Bring it in 250. That ain't enough, John. Said, at least we got money. Sarah, some people ain't got no money. I said, John, but it ain't enough. Said, at least we got some, John. Then we started arguing about something we didn't even have. And that was money. So he said, well, I've had enough. I'm leaving. I said, well, good riddance. So he left. I continue on working. Washing clothes. I said, Sarah, you got to do something, Sarah. But as time would have it, the World's Fair comes through St. Moon. All the big time colored movies and shakers was there. W.E.B. Jane. Booker T. Washington and his wife. Oh, Mary Church Charelle. Ida B. Wells. Oh, J.R. Cliff and Nanny Helen Burris. They all come through the World's Fair for the National Negro Economic Summit. Oh, they all got up and talked about how we can better ourselves and our race. But when it was time for Mrs. Booker T. Washington to speak when she got up. Oh, I looked at her. She had the most beautiful dress. No wrinkles, no holes, nothing. A string of pearls and a shoe shine and her hair's tucked in at the bottom. Her hair was so shiny it looked like she breasted a hundred times. I'm so busy washing clothes. I ain't had time to be washing my hair and taking care of it. It's all dry and brittle and falling out. But when it was time for her to speak, she went up to the front and she had that book learning. Oh, she says, ladies and gentlemen, it is so important to send our children to college and professional schools so that they can become doctors, lawyers, teachers, business owners and bankers. It's also important to send them off to, of course, trade schools so that they can become carpenters and cabinet makers and farmers. But mostly important than that, we should develop some kind of family business so that we can pass it on from generation to generation. And that way we can have our own wealth and develop economic independence. We won't have to depend on anyone for survival. Oh, I didn't hear no more. She said, start some kind of business so that we can develop economic independence. I said, Sarah, you got to do something. I went home and got my bowl, and I stir up everything I could. Salt, sugar, pepper, all my lime soap, molasses. I went outside, I got some red clay. I mixed it all up.  But I got to try it. I got to do something. So I put some on my hair. About a month, nothing happened. So I mixed it up, mixed it up again, nothing happened. I went to sleep and I had this dream. It was so real, all the ingredients I needed. I dreamt about it. I mixed it up and put it in there. I said, Sarah, here you go again. After about two months, I said, all right, Sarah, oh, let's see, Sarah, it should work by now. Let me look in the mirror. Thank you, Jesus. Thank you. My hair was this long. I tell you, this long, dry and brittle. All right. All right, Sarah, what you going to do? What you going to do? I said, all right. I think the other ladies might want some of this. So I put it in jars. After I finished my washing, I went out and I tried to sell some. Sarah, what you got to do? I'm going to put on my best white blouse.  That's what I'm going to do. So I put on my best white top and my black skirt, and I went out and I tried to sell three months and nothing happened, but I went out every day because I knew something was going to happen. And after about four months or later as well, I'll try something. My hair is not growing and it's falling out. I'll try. And after about a month, she came back. She said, Sarah, look at my hair. It's growing. What you have is good, Sarah. A lot of ladies want this, Sarah. I'll take three bottles. I said, thank you. Thank you. Oh, and the ladies were lined up at my door and I woke up in the morning. Everybody wanted Sarah. Hair growing tonic. I said, all right, I had to get four ladies to help me. I'll take north, you go south, east, west, we'll cover all of St. Louis to have everybody get something. The money was rolling in. I'll tell you the money was rolling in. And here comes Maggie Malone. This was the lady I used to help sell her hair bleeds.  Oh, I knew. I had to find some other place to go. My brother was in Denver, so he died a while back and he left his widow and two dogs. So I went to Denver, new territory. He says, all right, let's see what I can do here. When I got there, the ladies found out that I was Sarah from St. Louis. They all lined up, come to get Sarah's hair tonic. He says, oh, things are clicking tight. Money was rolling in. And then, of course, a gentleman I knew from St. Louis came up to Denver. His name was Charles Joseph Walker. Everybody called him CJ. CJ and I, of course, I knew one another, we knew one another back in St. Louis. Then we got to talking and pretty soon CJ and I got married. So I said, I don't want to call myself Mrs. Charles Joseph Walker, Mrs. CJ Walker. I want something that sounds important. I'm going to call myself Madam, Madam CJ Walker. That sounds important. So by now, Lila, of course, was out of college. She was a striking six-feet tall, beautiful, and she had that book learning like Mrs. Booker T. She became the spokesperson for my company. I said, all right, Sarah, it's time to form a company and get some people around you who know about the laws and the tax and everything. So I formed the Walker team. Says, now, of course, Lila became the spokesperson for the company. Ms. Alice Walker, of course, and Ms. Violet Davis Reynolds, I met these two women at a convention. They, of course, would help me with the office and all the paperwork. Ms. Alice Davis would take care of all the people who had mail or the catalog orders and keep all the papers and files and all the court documents. Ms. Violet Davis Reynolds would travel with me. She would keep track of all my speaking engagements and she would write those fancy speeches for me. Mr. Robert Lee Brockenbauer and Mr. Freeman B. Ransom, two of the best-colored lawyers in Denver. They knew all about the laws and the taxes. They would take care of the courts and, of course, the banks and anybody who wanted to sue the company, would take care of things like that. Take care of all the money coming in and out. And then, of course, myself and CJ, we formed the Walker team. We met every week. We discussed all the money coming in, going out, the territories we were in, the territories we need to conquer. All the monies were rolling in, I tell you. Things were clicking, things were clicking. Oh, I said, now, Madam CJ, it's time for you to enjoy some of this money. Lila certainly is enjoying the money. My daughter is wearing a turban hat with real diamonds and turquoise in it. Having lavish parties every weekend becoming the social life. Says Madam CJ, yep, it's time for you to enjoy some of this money. So, I started wearing silk dresses. I said, now, don't do too much, Madam CJ.  You can enjoy some of this money, Madam CJ. And, of course, the first thing I got was a string of pearls like Mrs. Boogie T. Washington. Oh, that was nice. Started wearing gold watches and gold bangles. Says, don't do too much, Madam CJ.  Started wearing diamonds. I said, now, look at this. Now, me and Lavinia didn't even have a good dress between the two of us. Here I am wearing diamonds. And, of course, Lila, of course, made sure that, why don't you have to have a pedicure and you have to have a facial and you have to have makeup on. So, I put on my makeup.  And, of course, I would wear hats with feathers on. If you didn't have any feathers on, I would say, put some feathers on it. When I go to a convention, they say, which one is Madam CJ? I said, see that one. That one that's wearing the feather. The feather in her hat. Oh, then I had fur. I had a black fur. I had a white fur. I had a gray fur. I said, if there's any other color, I want that too. So, now, Madam CJ, go on, Madam CJ. Now, I always tell my workers and I always tell my team, don't have a hole in your fist so tight. If you have the money, help the people. It seems like the more I gave away, the more it had come back to me. It started calling me a philanthropist. Because I gave them to the NAACP. I gave them Tuskegee Institute. I gave to the shelter homes and schools and other colleges, YMCA's. Oh, wherever I could, I just gave away the money. I gave away the money to the Color Women Association and helped them to get women to vote. Oh, and we read, of course, to New York to have the solid protest. Protesting, of course, lynching. Nobody said a word. So, it started calling me a social activist. I didn't consider myself a social activist. I just did whatever I could. Oh, of course, now it says now. What you going to do now, Madam CJ? I said, I want to do one more thing. One more thing I want to do. I want to build myself a mansion. Something somebody will come from miles around. Other citizens, states, and countries just to look at my mansion. And then, of course, we can have our conventions there. And, of course, I did.  I says, no. Anybody can come and have their meeting gear. This is not my house. This is the people's house. I just built this house to let people know that wherever you come from, that this can be done with hard work. It's not my house, it's for the people's house. Oh, so? It's time and all, of course. People would just call me to their conventions to speak, and CJ and I went across the country on the train with flyers to advertise the Madam CJ Walker Manufacturing Company. Oh, got real estate. Things were good. Things were clicking, I tell you. Things were rolling in. One last company note. I started my company with $1.50. Within a year, they had cultures, the women in their salons, that were making between $25 and $35 a week. That was unheard of. My salespeople were making $2 to $10 a week. Within a year, I employed a thousand salesperson bringing in well over $1,000 a month. Things were clicking, I tell you. Clicking. Within six years, 4,000 salesperson all over the United States, my factory employed $3,000. By 1918, bringing in well over $250,000 a year, on my way to making history and becoming the first black female self-made millionaire. So goes the life of the ride-to-riches story of the first black self-made millionaire, Madam CJ Walker, Sarah Breedlove, Mac Williams, Davis Walker, America's first self-made millionaire. Thank you. Thank you. Well, thank you so much, Dr. Sentry. That was fantastic. We learned so much about Madam CJ Walker in that really dynamic performance. And now we are very fortunate to have Ms. Lilia Bundles, who is an award-winning journalist with ABC and NBC. She is the great-great-granddaughter of Madam CJ Walker, and she is going to answer our questions for us this morning. So welcome and thank you, Ms. Bundles. And before we start asking the questions, I just want it because we do ask people where they're from. So just to let you know that we have folks from Illinois, from Washington, from Alabama, from Southern Maryland, from St. Louis, you know, where she lived, and from New York. And from Colorado in there as well, and Pennsylvania. So anyway, thank you so much for joining us. And so let me find my questions here. It says, Madam Walker still inspires entrepreneurs and philanthropists. Why do you think her legacy is so enduring? Good morning, Missy, and it's really great to be here. It was wonderful to see Dr. Dorothy's presentation. She really brings Madam Walker to life. So now we have an idea of what Madam Walker would have said herself. But I think Madam Walker is enduring because she is someone who really does embody the idea of working hard, the American dream as we, you know, believe it as a rags to riches. But really because she challenged some of the stereotypes and challenged some of the conventional wisdom as a woman. Before women had the right to vote, she pushed for suffrage, but she also created a business when women couldn't own property. And she, it seems to me she made these personal accomplishments. But what makes her enduring is that she created opportunities for other women. She helped other women become educated, become economically independent as business women. And then that made a difference from people whose mothers, grandmothers, great aunts had been Walker agents. And then who took this idea that she planted of being not just financially successful, but being active, politically active and civically active in her community. Thank, that's great. And so here's an interesting question. I understand that Alex Haley had an interest in Madam Walker's story during the 1980s, and that he became a mentor. Do you know anything about that? Does that ring a bell with you? Oh, absolutely. Alex Haley was a good friend and a mentor. I had written my my paper in graduate school journalism at Columbia about Madam Walker. And in 1976, that's when Roots was first on the scene as a book and then later a miniseries. And Alex approached me and my family about doing a miniseries and a book on Madam Walker. We had dinner in New York and he described what he was going to do. He was going to hire six or seven researchers and you know, of course he had done lots of research at the National Archives and I said at the end of dinner, I said, Mr. Haley, I'm already doing this research. I have access to all of these papers and documents that my family has saved that the company has saved. I'd be happy to do that research for you. And that really led to a friendship until he died in 1992. He never finished writing the book but we had many discussions at his farm in Tennessee. We talked often, we mapped out what the story was going to be. In the meantime, I wrote a young adult book different from the one that you that we're talking about today on a freighter trip with Alex in 1991 and then Alex died in 1992 and then I went on to write the four books that I've written since then but he definitely was a mentor with a great deal of interest in Madame Walker's story. Okay, here, these people who are watching, they are just so well informed and many people saw a self made and the fictional Netflix series it just came out inspired by on her own ground, your non-fiction biography you wrote about Madame Walker. What were some of the things you liked most about the series? Well, I think what I liked most about the series is that Octavia Spencer was spectacular in the role. Every time she came on screen, I really thought that she did a great job of embodying Madame Walker, her spirit, her tenacity, her courage. I loved the scene with the market women where she's trying to persuade them to become Walker agents and those things were important to me, that she built a business that showed the struggles of a woman building a business. There were a lot of things for people who care about the details and the facts of history that were just really historically inaccurate. The scene with Booker T. Washington really portrayed him as just a misogynist and while he was a man of his era that was inaccurate. The daughter, the character of the daughter of Alelia Walker was really a 180 degrees different from Alelia Walker who was really kind of a dignified person who actually had a great singing voice even though she's portrayed as not being able to carry it to who didn't have a girlfriend named Esther. So that was off. The other thing I think that is that really stands out is that the character Annie Malone who was Addie Monroe in the series in real life Madame competitor was Annie Malone but they both became really successful they both were philanthropists. Annie Malone was not a light skin woman so that whole colorism theme was entirely fabricated but they were real life competitors who both developed successful businesses. Oh, great. Okay, here's another one. Did she teach Madame CG Walker teach other people of color how to get a better life? Did Madame Walker do that? Yes. Yes, absolutely. But you know it's interesting I mean even the title self-made of the Netflix series is kind of the opposite of what really happened. I mean yes she was a self-made person in the sense that she didn't marry someone who was wealthier and didn't inherit the money. But I think that there's a term called mutually made that really describes Madame Walker's life when she was a poor washerwoman, Sarah Breedlove, McWilliams Davis she was really influenced by the women of St. Paul African Methodist Episcopal Church, the school teachers the women who were involved in the National Association of Club Women the Club Women's Movement women who were working on suffrage and it was those women in the missionary society in the choir where she's saying who began to give her a vision of herself as something other than an illiterate washerwoman. But that was happening with women all over the country. So Madame Walker, poor Sarah Breedlove took those lessons that she had learned from seeing women in their collective action. And then when she began to develop her business and began to recruit agents she realized that these women yes they wanted her hair care product which worked but they also needed education they needed economic independence. So that became as much the motivation for her as just selling hair care products. So she was making it possible for other women who otherwise would have been working as maids and laundresses and farm workers to have some economic independence. Okay, another one. So when did you begin to discover the stories of your great great grandmother and your great grandmother the both of them because I'm almost finished with a biography on Aaliyah Walker and her life in Harlem during the 20s. But you know I really began to discover the story of these women before I could see my mother and I would visit my grandfather in the apartment where he and my grandmother had lived. My grandmother died in 1945 and in the mid 1950s my mother and I would visit my grandfather in the apartment and in my grandmother's room really had not been touched since her death in that decade and in the dresser drawers were things that had belonged to Madame Walker and Aaliyah Walker into my grandmother. Ostrich Feather fans, mother of pearl opera glasses, Madame Walker's shawl and then the silverware that we use every day at home had Madame Walker's monogram MCJ CJW on it and my mother was Vice President of the Madame CJ Walker Manufacturing Company so I would go with my mother to her office in Indianapolis and play on her typewriter before there were computers and on her adding machine before there were calculators visit the ladies who worked in the factory who were still hand mixing Walker products so I began to learn about this story long before I understood its significance and then fast forward much later I began to become interested in Aaliyah Walker and her life during the Harlem Renaissance we had first edition books by Langston Hughes and Count T. Cullen and then in graduate school I really began to seriously research Madame Walker's life and discover these other dimensions about her so a related question do you still have all those personal items that you mentioned her ostrich feather fan, her hats her jewelry and are they in a museum or on display anywhere I actually have a quite a collection here in my house my Madame Walker family archives so yes I have some of the clothes I have letters, photographs a few pieces of furniture jewelry some other things books but we also have donated things to the National Museum of African American History and Culture at the Smithsonian and I lend things to different museum exhibits but the largest collection of items of Madame Walker's are at the Indiana Historical Society where we donated in the early 80's thousands of papers photographs and business records that had been Madame Walker's and the Madame CJ Walker Manufacturing Company and there is an exhibit that's up right now at Indianahistory.org and the collection is digitized okay so switch here many of us are doing genealogical research about our families any advice on how to organize the material well you know with the National Archives that's one of the very first places to get started to do research to use the NARA documents that are on ancestry but then to do your own visiting when things do open back up I really benefited from my friendship with Alex Haley when I started to do the research for the project that he had hoped to do he gave me a really key piece of advice he said to make a manila folder for each year of the person's life so for Madame Walker born in 1867 I made a folder 1867 and then she died in 1919 so I had a folder for every year and then I expanded it to Amelia Walker's life and then going backwards to her parents lives and I began to put things in that folder I've been doing that now for more than 40 years so I have now two rooms of my house that are the documents but the point is is that it's really important to organize your material because over time you accumulate so much you don't remember what you have so in addition to those chronological folders I also have folders for all of the key people in her life I will drop something in there for the main organizations for certain time periods certain eras it is really important to keep the material organized so that kind of leads into the next question so what do you see yourself working on next what's your next project with all that material that you have so I've been working on a biography of Amelia Walker for really quite a long time I'm almost finished with it and it should be out next year and you know honestly once I finish that book I'm not writing anymore books that really require a huge amount of historical detail because these biographies writing about people who really have never been written about before requires a great deal of meticulous research that takes a lot of time so I still will be writing writing articles, writing poetry you know doing other kinds of things but this will be the last major biography that I write that really requires this level of detail because I want to have some fun and I want to read all the books that are in my house well this question is actually from me with all of the had anyone done a lot of research on Madame C.J. Walker before you started were you able to find any or were you really doing that original research no I really did the original research you know it's really surprising but when I wrote my young adult book that was published in part of the Chelsea House series in 1991 there had never been a biography about Madame Walker so the first serious comprehensive sort of comprehensive biography of Madame Walker was really a young adult biography and then I use those documents that are at the Indiana Historical Society this was when they we first before the collection was processed I use that I traveled to a dozen different cities going through courthouses historical societies public libraries the papers of other people with whom Madame Walker had any interaction old newspapers city directories so I really did the original primary research so in on her own ground the end notes are really detailed and I did that specifically to try to create some bread crumbs for scholars who would come behind me you know at this point it's very different doing research I go to a library go through microfilm to read old newspapers if I wanted to read the Chicago Defender I had to go to Chicago you have to go to the National Archives to go through microfilm to go through census records now those things are digitized and you can do that you can do that work online but I'm really happy to say that some of those bread crumbs that I put out there for other scholars have been used by people like my good friend Tyrone McKinley Freeman who's recently written a book really focused on Madame Walker's philanthropy called Madame Walker's Gospel and Giving and other scholars like Tiffany Gill and Nalihwe Brooks and others who I have really enjoyed helping as they were writing about Madame Walker and using some of that original primary source material to then go on and expand on that in their own creative ways thank you another question a little bit different area did Madame CJ Walker have any patents on her products so there are trademarks that are at the patent office where she is really the system her Walker system there are some things if one wants to do the USPTO search to find things that the Madame Walker company has a trademark and the names of some of her original products are trademark okay and then this interesting question so today we think of people who are starting a business that get financing from other places did she get any financing from banks or it was all her own yeah it was all her own the world is very different though even now you know women and people of color in an infinitesimal amount of venture capital money so there's still not a lot of capital available to women and people of color who are starting businesses and that is you know just really kind of obscene how uneven that distribution is but when Madame Walker was starting out she did approach a few people to try to sell some stock and she didn't have really very many takers so it was really her sweat equity she would make some money invest that money back into the company and you know over time the banks that you know weren't willing to lend to her over time they saw that she was making such large deposits that they did you know extend her a line of credit but she really was self financing okay and then we really were out of time thank you so much we have one more question for you if Madame live today what advice would she give to our youth absolutely she would say education is really important she did not have an opportunity for much formal education as a young person she did she ended up hiring a tutor once she had the money and the means to do that but she also gave scholarships to a number of schools to Mary McLeod Bethans Cookman College to Tuskegee Institute to Lucy Laney's Institute and several other schools so she really believed in the importance of education and she would say it's really important to study hard and then to use that the brain power that you develop if you go into business if whatever you're doing make sure that you're giving back to the community that you are being a philanthropist and and supporting your community and being politically active well thank you so much and thank you Dr. Sentry I'll come back on again to say thank you so very much to both of you this was an amazing program and I know that all of our viewers got so much out of it we did have I saw a few more from New Hampshire and other places so we are going to now put up the slide as I promised for the DoxTeach activity and then we're going to put up the slide for the so you can purchase the book about Madam C.J. Walker so again thank you all so much and have a wonderful rest of your day


I'm taking victory lap for the Army Navy game. But the game is tomorrow. Tomorrow? Well, I'm getting the early start. Let's give it up for Army B Navy.


Oh, yes, he's ready. Hey, thank you for introduction. I'll talk about... Stop. Can I start? Sure, please start. Okay, so I'll talk about Lossy Compression of Matrices by Rockbox Optimization. Mix Integer, Non-Linear Programming. I'm Tadashi Kadowaki from Tenso, and this work is a joint work with Mitsura Anbei. So Lossy Matrix Compression, or Metrics. The composition is a task that the target matrix is divided into the product of the matrix, integer matrix, M and real matrix, C. So we want to minimize this the difference between two matrices in terms of M integer and real matrices. So this is Mix Integer, Non-Linear Programming, and it's NP-hard. And application, this matrix, the composition, we demonstrated the image recognition as well as voice recognition. For image recognition, the memory size is one-third, and the execution time is one over thirty-seven, which is good for edge computing. So how to solve this matrix, the composition? The most popular approach is, for example, it's already demonstrated by Yoon, and his colleague in actually last year, they optimized the real variables first after that, then optimized integer bytes and repeat these until it converges. And one more also demonstrated another approach, which is one-rank approximation, the outer product of two vectors, make a matrix, but it's poor approximation, so we can repeat this one-rank approximation to the integer of the matrix. Again, to have good approximation. Here in this presentation, we propose another approach by data-driven approach, and if we generate first the relationship between matrix and cost, and if we have a dataset, we model the relationship of the data. And once we have a model, we can optimize that model. And the optimization finds new candidate and we can repeat this, and the dataset is growing, and we have better modeling and we have better optimization. So the first step we want to remove continuous barrier, and that means we translate, convert the mixed integer programming to integer programming. So if the M is fixed, C, the matrix C is calculated like this, and instituting this relationship, we have P of M. So now we have the optimization of M with this non-linear integer programming. So again, this is what we want to do. However, the cost function here is very complicated. So we forget it. I mean that we take a data-driven approach. So we don't directly optimize this complicated function, but we generate data using this function. Again, we input the matrix M and we have a dataset and we model the relationship of the data matrix M and cost of the data matrix. And we model in the cubo form this case so that we can optimize the model using Ising Solva. So Ising Solva outputs the next candidate to be calculated cost. And repeat again, so we will have a better solution. Okay, so we have a general proposal of black box optimization for binary variables. One is the VOCS proposed by Battista and Procek. And FMQA is proposed by Tai and his colleagues and it's presented in AQC. The difference between two algorithms is the model generation. So VOCS utilize Beijing inference to infer the distribution of cubo. But we don't have the technique to directly optimize the distribution of cubo so we need a specific cubo matrix. So we sample from this distribution. So it's kind of Thompson sampling and these algorithms randomize the algorithm. For FMQA it utilize factorization machine so it's point estimation and deterministic algorithm. And I'd like to mention that the VOCS QA version of VOCS is also reported in Koshikawa and colleagues in AQC. Okay, so there are several potential variations of these algorithms because VOCS has prior. The original prior is a horseshoe prior vanilla VOCS we consider to another additional prior, normal prior and normal gamma prior. Those have hyperparameters so I optimized it before. And FMQA it has a hyperparameter K associated with the dimension of related vectors. So I applied KQ8 and 12. And we also test random sampling. For easing solvers in addition to the conventional Schmittet annealing as well as quantum annealing we test the Schmittet quenching which is the quenching the temperature to zero immediately. Before moving to the results I'd like to share energy landscape of this problem. So the product and the so in variant and the permutation columns like this, the order is changed or sub-signs expressed in the cartoon. So there are in this case 48 4K or 3 these 48 solution is spread here and this is a clustering analysis of those solutions. And I cut here into the four groups which we used in the right analysis and as this is degenerated the data augmentation can be applied. So this is a result of the algorithm comparison. So among those algorithms you see first this is iteration of step and this is the best result error obtained during the iteration. So you see the vanilla B O C S and normal flyer B O C S so the best performance. So I tested 10 random instances and summarize in the table. And you see the end B O C S which is the best success rate is 36% for the solution accuracy and for execution time it also fast compared to other algorithms. Okay now the data augmentation. You see the data augmentation does not improve the results because the QA data fitting QA model can't approximate the cost function globally that causes this poor performance. And next the comparison among using solvers you see a significant difference among using solvers including quenching the table again. So I add this part so you see a tiny link with flyer B O C S and QA version SQ version and you see the best performance among all algorithms I think there's no significant difference between 3D algorithms. However you see the execution time QA takes a long time because QPU time itself is very short but you see the very large overhead as Catherine explained in the first day. Okay the final result is that so I visualize the relationship between the balance between exploration and exploitation among four groups. So in the random sampling there are four colored lines almost identical that means it's completely biased towards exploration a similar figure we see in the data augmentation algorithm so that means it's the poor performance and vanilla B O C S also tend to be exploit. Okay other side so for FMQA this is extreme case the early period it biased towards exploitation so it just focusing on very small area of the solution space in the early time so it biased towards exploration and the best 3D performance algorithm it's well balanced between exploration and exploitation. Okay summarize the results of the the mixed integer non-linear program is transferred into the non-linear integer program program and solved by binary B O and the normal prior shows the best performance. We have some insights for further development the first one is the data to make the dataset cover globally in such space however we simplify the fitting model it can't approximate the cost function globally and the second one is that there's no significant difference using solvers this is non-trivial because SQ is greedy optimization the surrogate model might be easy compared to the express form of the original cost function visualization of the balance between exploration and exploitation help our understanding. Finally the advantage of quantum annealing in the binary black box of my decision that's it thank you for your attention questions and comments thanks for the talk I'm a bit confused on why the documentation makes the result worse I mean you are kind of encoding the symmetries so if it doesn't help I would have expected not to change the result but not making it worse could you comment on that yes that is related to this and that is my thought at this time there's no evidence but the strengths of the black box optimization with relatively simple modeling is that we first explored the large space solution space without bias but if we get some biased data sampling and it's around global minimum basin we should focus on the very small area and intend to get samples and we get fine tuned cost function representation on this area and we will get good results but it's very difficult to fit with Cuba very limited to explore the function to fit global with many global minimum that type of structure cost function it's very difficult that's why in this case the documentation doesn't work well does it make sense so is it like those states are separated by I don't know they are far away or separated by barriers or something I still don't get it yeah that's about I don't study well but the structure the reason why we have many degeneracy is like you see these vectors constructing one matrix and if you change the order of this case the red and blue change and we also change the order of red and blue here that is equivalent so this causes a very complicated landscape cost function and I have no idea how hard the cost landscape looks like but that is a future study I should contact thanks I have a question did you so you you transform the original program to the another program and how what is the accuracy of the solution obtained by new formulation okay so again the tubal data fitting is not perfect but in this case this is the residual error that means this line is the second best so below this line this is among the best but if the curve touch this line that means some of the runs it's the exact solution so that means this approach is not but it's pretty good approach I think okay thank you are the questions if not let's thank you