Okay, so Stefano and Zé and Matos, do you think it's a good time to start? Yes. Yeah, go ahead. Okay, very good. So welcome everybody. Sorry for the late start. We are five minutes late, but no problem. So let me first you make an announcement and actually it is related to the organization of this school and workshop. So as you know, ICTP is one of its goals is to foster mathematics in developing countries of which Brazil is part of. So I discussed a lot with Stefano Luzato ways to try to make this development, especially in the region that I come from and I work, which is in the northeast of Brazil. And one of the ways that we found was exactly by making this event. And another way was trying to select people for postdoctoral positions that could work here and also have transit at ICTP. So I would like to advertise a postdoctoral position. So I'm sending here the link that is paid that is funded by Instituto Serrapilheira. And this postdoctoral position is for people in dynamics here in my university. And they will also be allowed to have an extended visit at ICTP. As a matter of fact, the link that I sent you has two jobs. One of them is in dynamics funded by Serrapilheira. And there is another one in differential geometry also funded by Serrapilheira. But I should focus that the extended visit is only intended for those applying for the dynamics position. Okay, so you can find further details in this link. So another above on top of having this postdoctoral position, we also decided to have this school and workshop on the field of non-uniform hyperboleicity, which is the field that we both work and on which dynamics is focused in a lot in the past years. So there are two important techniques in order to understand non-uniform hyperboleicity. And they are Markov partitions and young towers. Markov partitions, they were developed in the dynamical perspective in the late 60s and early 70s of last century by Sinai, Adelaide, Weiss, Ratner, Bowen, and others. And young towers, they were introduced with success by Young in the 90s and then later developed by Chernov and other people. So our idea was exactly to try to present these two techniques and their developments, their very recent developments to allow students, postdoctors, and also professors to learn these two tools and to compare them. So in the past three weeks, we had two mini courses, one given by myself and another given by Jose Alvis, which was, I presented the Markov partitions perspective and Jose Alvis presented the young towers perspective. And everything was recorded so you can find on YouTube all the videos. So this was the part of the school on the event. And now we arrive at the part of the workshop. So this workshop will finish the activities of this event. We will always meet in this week at the same time slot, which in Italy is from 2 p.m. to 4 30 p.m. in Brazil, on which many people are in the audience is from 10 p.m. to 12 30 10 a.m. Sorry, to 12 30 p.m. And we will have two talks every day. And today we will have Carlos Matheus and Yakov passing. So it is a pleasure to to introduce in Carlos Matheus, which works at CNRS and they call it technique. And Matheus will talk about young towers on non positively curved surfaces. Thank you, Matheus. Yeah, thank you, Yuri, Stefano and Zé for organizing the event and inviting me to talk here. So it was a lot of fun following the mini courses on YouTube. And again, I hope you have fun also seeing my talk. So yeah, as Yuri said, I'm going to talk about the aspect related to this course, which is young towers and the key of correlations in the specific case of non positively covered surfaces and everything that I'm going to talk about is trying to work with Yuri Liman, Ian Melbourne. So before entering into the subject, let me quickly show you the plan for the talk. So the plan is to recall you something about what is known about ergodic fury in negative curvature. So it's classical top, but it's always good to refresh your memory. Then I will start talking about non positive curvature for surfaces and what was known prior to our result, which is basically a result of constructing young towers and obtaining statistical laws in particular the key of correlations and central limit theorems for a particular class of non positively covered surface. And then in the final part of the talk, I'm going to explain the proof of the result. Not everything because there are many aspects to this proof, but at least some what we could call the main ideas. I don't know. Yeah, so that's the start. So for me, S will be always a Riemannian manifold. So finite volume always. I'm not considering it's volume here. Yeah, dimension at least two and coming with a Riemannian metric. And I'm doing that to not mix up the manifold with the phase space of the jedezik flow. So GT for me will be the jedezik flow, which is a flow taking place in the unit tangent bundle. Or cotangent bundle depends on your preference. And then just to make things clear, I'm going to denote by me deal with measure. So the lab bag measure on these Riemannian manifold, which is invariant under this flow. And I'm going to call M for the phase space of the jedezik flow. So the unit tangent bundle. So as for me, it'll be the surface. But as everybody knows, the jedezik flows take place really on the unit tangent bundle. So takes place on M. Okay. Now, what people, well, the history of analyzing ergodic properties of jedezik flows has a long history. So of course, we should pay tribute to the work of Hopf in 39. And Hopf in a similar work, he introduced what people call today the Hopf's argument to prove that the jedezik flows on surface of compact surfaces of negative curvature. They are ergodic. And as I told you, this is what people now call Hopf's argument. And the idea is that basically he studied a big of average along invariant manifolds. So he noticed that basically forward big of invariant average are constant along stable manifolds. And similarly for past big of average along unstable manifolds. And then he used a kind of Fubini streak, which was possible because the foliation, the invariant foliations are C1 in this setting. And so by combining these two arguments, he was able to prove ergodic so that functions which are invariant are constant almost everywhere. But this property of having C1 invariant foliations is very strong. And in particular, it was discovered by a Nozov that it almost never works in higher dimensions, in some sense. But in any case, several years later, so in his thesis in 69, a Nozov noticed that actually to run out the second part of the argument, you don't need the full strength of C1 in regularity. You just need to know that the foliations are absolutely continuous. So to apply this Fubini argument, just need this weaker property, which hopefully, which luckily holds in our setting. So thanks to dynamics, the foliations which are usually just holder, they are holder plus this extra absolute continuity property, which is sufficient to conclude. So yeah, thanks to dynamics. We don't have arbitrary holder foliations, but a special kind of foliations. And interestingly enough, another regularity discussion come up in our setting when we discussed our work with Iurian member. But it's not exactly this kind of regularity, but it's interesting that there is always a kind of regularity discussion that we should make when talking about these statistical properties. But this is the context in negative curvature. Next, once we know that the flows are ergodic and actually mixing, it's not hard to see, you can ask about the speed of mixing. So in other words, you look at the correlation functions. So for instance, you could think of these functions here, phi and psi to be characteristic functions. And then this would be just the probability that mu, the probability that the intersection of a and gtb minus the probability of a and probability of b goes to zero. So basically how fast these things correlate when you apply the flow to one of the sets. And these speed well, it goes to zero because of the mixing property. And then you can ask for rates of decay, at least if you pick observables which are smooth. I mean, it's known that if you pick arbitrary observables, then even the best chaotic systems have slow decay for arbitrary observables. We can always cook up very tricky observables making this very slow. But if you look at, say, C1 or C infinity observables, then there is a hope to pick up some speed here. And actually, you can ask, well, why you are interested in these questions of the key of correlation as well. It's a natural mathematical questions once you see mixing. But not only that, it has applications outside dynamics. So in particular, it was used by asking McMullen in the 90s to count integral points in algebraic varieties, special types of algebraic varieties. So in other words, you can use exponential mixing to count things. It was used in 98. So asking McMullen, I think it was from 93. Klein-Bock-Margulies is from 98. They also used that to solve problems of, I mean, diophantine problems, trying to pick up diophantine vectors inside analytic manifolds and the method to pick up these diophantine properties along manifolds was to use exponential mixing and distribution properties. And more recently, in 2012, if I remember correctly, exponential mixing for the frame flow was used to build surface subgroups inside hyperbolic, inside the pi1 of the mental group of three manifolds. And this is the work of Ken and Marcovic. And the list continues. But I'm not going to mention that. So the message here is that rates of mixing are important not only for, say, intrinsic beauty of mathematics, but also has applications through number theory and geometry and so forth. Okay, but let's get back to mixing. I'm not going to mention the application of mixing, but just how to prove some results. And in this direction, I want to say that in 87, Marina Hatner got a very precise exponential mixing result, but in the case of hyperbolic surface, so constant negative curvature. And her idea was to use what people call a Bargman's classification of unit representation of SHR. So basically the idea is that from the spectral gap that we get on the laplacian of the surface, you can convert that into isolation property from the trivial representation. And this basically converts into the key of correlations. But of course, this theory is very algebraic, so uses representation theory of SHR. And so there is little chance that this kind of argument extend to variable negative curvature. That's the point. But much later, Chernov was developing a method which he called the time Markov approximations, which actually was he started developing these methods together with Bonimovic and Sniper billiards, but then he wrote in 98, a paper on geodesic flows. And he introduced this technology of what he called Markov approximations, which was a kind of pre-Yangtaber somehow, if you wish. And then a condition called non-uniform integrability of the foliations to derive a stretched exponential mixing for certain geodesic flows in negative curvature. Okay, so a stretched means that this is the case at not exponential rate, but exponential minus the square root of t, basically. Okay. And then famously, Dogopiat was able to go from stretched exponential to true exponential using what people call today Dogopiat estimates, which you can summarize in one phrase. I mean, of course, this is a kind of disrespectful to Dogopiat, because I'm trying to summarize this beautiful work in one phrase. But if I had to do that, I would say that it's applying Von Degpore put lemma to temporal function. So basically, he wants to pick up oscillations in the Laplace transform of the correlation functions. So you look at this correlation here as a function of time, you apply Laplace transform, and then you try to get cancellations in the kind of Fourier or harder Laplace expansion. And to get this cancellation, you apply this idea of that the non-integrability translates into a beautiful oscillation along temporal functions. And you use a classical lemma, due to Von Degpore put telling when exponential integrals or sums exhibit cancellations. And then, of course, once we got this result for exponential mixing for a class of systems, then everybody got excited and there was a huge literature, what was developed after that. I'm not going to mention everybody, so I'm sorry if I'm not mentioning your work today, but I would like to mention just a few names on this direction, because the literature is huge. But let me mention that Leverani was able to extend Dogopiat's work for another contact, another flows. And more recently, Suzy and Jiang announced a solution to this so-called bow and well conjecture, saying that if you have in three dimensions this time, so you have a compact three-dimensional manifold, you have another flow on it. And then you can improve topological mixing into exponential mixing. So it's kind of funny that topological property implies into a quantitative property. And this is for all equilibrium measures for holder potential. So maximum entropy measures, Liouville measures, yeah, all of them, all interesting measures. Okay, so this is the scenario for negative curvature. And now I'm going to add some zero curvature in my manifold. And so again, the setup is S now for me to be a compact surface with non-negative curvature. M is the phase space of the geodesic flow. Mu is the Liouville measure for me. Gt is the geodesic flow, sorry. And then I'm going to denote, I've been given a vector. I'm going to denote gamma xt, the geodesic generated by this direction. And then I'm going to define the degenerate set to be the set of vectors generating geodesics along which the curvature vanishes identically, okay. And then the regular set will be the complement of that. Now, I want to tell you why a gadget called Jacobi Fields are interesting when talking about geodesic flows. So you probably know this, but let me remind quickly what is the rationale behind these considerations. So the idea is the following. What is a Jacobi Fields? So a Jacobi Fields is you have a geodesic and then you have vectors along this geodesic, which are these vectors in Gt. And then it's a vector field satisfying this equation here. And so it's a little bit bizarre if you present it like that, a better way is to say the following. You start with a geodesic and now you consider one parameter variations of the geodesic by other geodesics. Okay. So now I have two parameters, I have a family which is gamma t comma s. Okay. So t is the parameter along the geodesic and s is another geodesic. And now if you derive the variation at each point of the initial geodesic, you get a vector field and the vector field because of the condition that the variation is by other geodesics satisfies this equation. So that's the point. This equation here characterize the fact that you are doing variations by other geodesics. So in other words, what you are trying to do is understand how geodesics are behaving nearby, given one that you are interested in. In particular, if you use these variations of geodesics to parameterize the tangent space to the unit tangent model, then you can do it. You can make an identification of, I mean, a variation. I mean, by definition, a tangent vector for the geodesic is a variation. And you think there's more variation of the geodesic, which we saw that it's identified with these Jacobi fields. And the advantage is that in these coordinates, the derivative of the geodesic flow is very simple. It's just, it starts at the initial point, zero, and then you just flow for time t. And you look at what fields you got. Okay. So these are coordinates which make the analysis of the geodesic flow very easy.  And since it's a remuner manifold, the unit tangent model is also a remuner manifold. It carries a metric, which is called the sacchymetry, which is very simple. And then I'm going to compute what happens to sizes of vectors in a positive derivative with respect to this metric. And if I do that, well, I'm going to do that along two subspaces, which are the stable and stable subspaces, which by definition are the initial points. I mean, this is a second order of the, so to give, to determine a solution, any initial position on the velocity. So this is why I'm putting G and G prime. So given this guy, I have a Jacobi field. And now I'm looking at, so the stable space is the space of Jacobi fields, which never, which stays bounded when I go forward. And the unstable one is the space of Jacobi fields, which stay bounded going backwards. And these spaces, they are nice because you can check by this simple formula that they are invariant on the digitizing flow. I mean, if I'm bounded forward, if I just go a little bit to keep this property. In our situation of non-positive curvature, they are one-dimensional. And they depend continuously on the point. And more importantly, they coincide if and only if, so I see a tangency between these spaces, if and only if I'm sitting on the degenerate set. Otherwise, they are perfect splitting if I add the flow direction. Okay. Moreover, I can quickly compute, as I told you, the derivative of the digitizing flow along these spaces with respect to the Sasaki metric. And then I get, well, after some computation, I get some function here, which I wrote the basic problem. So there are functions u plus minus, which are continuous functions, which vanish only at the degenerate set. And by compactness of my surface, this term here is bounded. So if you forget about this bounded term, what you see is just exponential of the integral of this function along a geodesic trajectory. So if this function is positive, so if you are away from the degenerate set, this function has a lower bound. And in particular, you are integrating exponential of some constant times t, at least. So you see exponential growth along, yes, if you are away from that. Similarly, this function is negative away from that. So you see exponential contraction along the stable direction. So in other words, the simple calculation shows that this space are really deserved to be called stable and stable subspaces. Because if you are away from back, you are really seeing exponential contraction and expansion. The only problem is that when you get close to that, these directions tend to close up. You get consciences. Okay. So this is the situation. And now let me remind you that there is a beautiful application of Passing's theory by Passing himself from 77, which explores these features of these bundles to show the following result that if the measure of the degenerate set is zero, with respect to the Liouville measure, then the degenerate flow is ergodic. Okay. This is basically one of, I mean, an important application of the theory of a non-informal hyperbolic system developed by Passing. Much later, Knieper considered other measures. So in particular, he considered the measure of maximal entropy. And he proved that there is a unique measure of maximal entropy in my context, if I'm rank one. In this context, it just means that the regular set is non-empty. Because I'm talking about surfaces. And more recently, even more recently, there was a series of works by Le Drapier, Lim and Sarig on one hand, and Bern, Bern, Skleeman, Haga, Fischer and Thompson, establishing many results about these types of measures of, I mean, equilibrium states in general. So we start establishing particular uniqueness and Bernolicity, which is kind of the top, the strongest abstract ergodic theoretical property that you can ask for in the chain of ergodistic mixing, K property, and Bernolicity. Bernolicity is kind of the strongest one, and they could establish these results in this context. For certain equilibrium states, I'm not going to give precise statements here, but I recommend reading the papers for precise statements. But in all in all, these results, yeah, so the advance, of course, the knowledge about statistical properties of these flows, but once you get to this state, then you can ask the same question that people ask at the end, another proof of mixing, which is, what about the key of correlations for these flows? And, well, this was not so well understood. And so the idea of the next result, which is, as I told you, I'm trying to work with Yuri Lim and you remember, is to offer some polynomial mixing statement force for a certain class of surfaces with no positive coverage. So the result is the following.  So this means that you're taking the profile on the plane, y equals one plus x to the power four in an interval. So you draw the graph, and now you make a revolution about the axis. And now you glue to make the surface boundary less. You glue two negatively curved surfaces on the boundaries to close up the picture. And then this is my surface. So this surface, I'm going to explain to you in a moment that it has negative curvature everywhere, except at the geodesic in the middle, corresponding to the revolution at the parameter x equals to zero. Okay. And so this surface has curvature, which is non-zero except a long one single close at geodesic. And in this context, what we can prove is just to visualize, this is like two balls that are glued in a point. No, I'm going to show a picture later. It's like two tori. Yeah, it's like two tori because I want to have in the end at least genus two. Yeah, because you start getting hyperbolic matrix in the torus. Could you show the picture now so that we understand? Yeah, I can show the picture now. Yeah, before engaging the results. Let me show the picture now. Yeah, so this is the picture. So this is the profile of revolution that I told you. I mean, this is supposed to be the curve one plus x to the power four, and then there is the x here. I'm rotating about it. And then I'm going to things of negative curvature just to cap it and make something with dot boundary. Okay. So this is the kind of surface that we can treat. Yeah, recently we think we can treat more things, but let me just keep to a simple statement. So let's take this example, which is very concrete. And then the result is that in this context, the presence of that single geodesic along which you have zero curvature makes the rate of mixing to drop from exponential to polynomial with a rate which depends on the profile. Okay, so it's this number. So the precise relation between the exponent in the profile and the rate of mixing is given by this function r plus two over r minus two. Okay. But in particular, since I'm right between four and infinity, I'm getting something between one and three. Okay, so I get cubic pk at r equals to four. And if r increases, my profile gets flatter and flatter somehow. And this speed gets closer and closer to linear. That's the morality here. And there is a work in progress by brewing Melbourne and tertiary saying that this rate should be almost optimal in the sense that in the same context, there should be, it should be possible to construct observables whose correlations are not very far from this rate up to some logarithmic term. But this work, as far as I know, it's not on archive yet. But yeah, I heard from Ian that this work is in progress and exists. And so presumably, the optimal, I mean, the exact rate should be about this thing here, maybe with some logs, some little powers, I don't know, but about that. Okay, so this is the result. Another result that we can get from the same, from the same arguments is a central limit theorem. So I'm just recalling what it means is conversing distribution to a normal distribution of this kind of average when I normalize by the square root. We also can get weak invariance principles, etc. Okay, so I'm not going to talk about all these results, but the idea that the same machinery gives everything. And of course, if you followed the course by Z, you know that all these statistical laws are going to follow from a nice young tower once we get the hands on a nice young tower. And so that's what we are going to discuss now. It's how to get young towers for this geodesic flow in this context. And you can believe me, once we have that, it's not hard to get polynomial mixing, central limit theorem, etc. Okay. Okay, so are there any questions? Okay, so if not, let me advance. So let me try to explain what we are going to do to get this young tower. So before that, let me compare with the situation for billiards. So actually it's funny because there is a paper from 2005 by Chernoff and Zang where they kind of consider almost the same scenario, but for billiards. So the idea is the following. You start with a dispersant billiard, but then you modify one of the obstacles to have a profile, which is exactly one plus X to the power R.  And so this makes a period of period two, where you have this kind of neutral behavior because the curvature is going to zero. And they could prove things for the map. But as far as I understand, you can have results for the semi flow, but not for the flow. Okay, so the analogous results for billiards are open as far as I know. I mean, at least that's what was said in a paper from by Melbourne in 2018. And I think the situation has not changed since then. But yeah, it's funny because for the same geometry for billiards, we know some things, but not everything. Yeah, but for the geodesic flows, we could do something. Okay, so let me get to the test shows. This is the picture of the surface. And as I told you, it's not hard to see that. I mean, this geodesic here for this profile has the Gaussian curvature is zero. So actually, I have formulas for these things. So you can check, for instance, these formulas in the book, it's a very nice book on differential geometry. And so we can look at the chapter on surface of revolution. And so what the car most says is that if you have a surface, which is the evolution of some profile function psi. And then you put cosine theta sine theta to do the revolution. Then the curvature is going to be is going to be given by this formula here. So it has some second derivatives in the numerator and then something which is not zero in the denominator. And in particular, you see that, well, the function one plus x to the power of four has the derivative zero at x equals to zero. And so the curvature is exactly there, but not nearby. Also, another thing that we're going to use is Clairro's coordinates. So there is a very nice remark by Clairro when people was trying to study geodesics on Earth, which is a remark that on surface of revolutions, you have this meridians, which are the circles. And then the remark by Clairro is that when you look at the geodesics moving on the surface and you measure the angles with respect to the meridians, then the profile times the cosine of the angle is a constant of motion. So it's an integral of motion. And so this is particularly useful for us because it allows to reduce easily a second order differential equation, which is the equation for geodesics, into a pair of first order equations, which are easier to estimate. And this is why we chose these examples first because at some point we need very precise estimates. I'm going to tell where. But the idea is that the analysis of the Clairro's equations, we can decompose well, we can first check that the dag set is just that geodesic in the middle, the curvature venison only there. And also that when you are trying to understand geodesics, which enter this kind of neck, which is the region that where I have this evolution profile, I'm going to call it neck. When you enter the neck there, you can do three things. Either you can aspire and converge to the geodesic in the middle. So I'm going to call this profile asymptotic. So I'm in the stable or unstable manifold of this closed geodesic.  Or you can cross, you spiral across the middle geodesic and go out by the other side. So I'm going to show you. And these three behaviors correspond to a certain value of the Clairro constant. Okay. So these are the pictures.  And so now what I'm going to do is the following. I'm not going to construct myself the young tower. I'm going to take advantage of the fact that the turnoff and co-authors of Markarian and Jeng notably working on a set of geometric conditions, which I'm going to call today turnoff axioms, which is a set of geometric axioms that allows you to ensure the existence of young towers with exponential tails. And then once we get that, we can take advantage of the theory of young towers and conclude this statistical law that I announced. So the key idea here is really to stick to turnoffs axioms, which are a kind of geometric gadget allowing you to ensure the existence of young towers. So he worked hard to show that these axioms implies the existence of young towers. So I'm not going to reinvent the wheel. I'm just going to use these results. And so what are these axioms? These axioms are basically, yeah, so, yes? Sorry, Matheus. Can I ask you another question? So at which stage do you go from the energetic properties of the young tower to the flow? I'm going in the very end because actually the idea is that I'm going to construct a nice section to be able to check for the map first. And then I have to analyze what happens to the return time to the section. And it's then that I'm going to get a polynomial mixing. It's because of the returns. So the moment everything you're talking about is you're going to, so when you say channels axioms, these axioms for the flow, for the map, for the map. They are axioms for a map. No, they're axioms too that will give the existence of the map, but the axioms on the flow or the axioms? Well, they are axioms on a given map. I mean, you have a map from a surface and then it's a list of eight axioms. I'm going to show some of them. And then if you check those axioms for a surface map, I mean, it's not to be a surface map. Then this guy has a young tower with exponential tails. And then there is the problem of passing this information to the suspension. And then we have to discuss, of course, return times to the section. And in our case, this part of the return time is not very difficult because we are dealing with contact flows. So the kind of only condition that appears in dogopiat's work and turnoff's work is automatic here. So this is why I'm not talking too much about the return time function because we are contact flow. So because we are geodesic, so yeah, the kind of uniform non-integrability affiliations are kind of automatic in our setting. Yeah, but I'm going to mention these two things in the end, I hope. Yeah, but the idea that we are going first to construct a very nice map and then try to check these axioms and then understand the return times. So what do we do? Yeah, so as I was telling you, we don't need to check too much. We don't need to care too much about the suspension function because we are in a flow, in a setting where the flow is very nice as content. Okay, so what are these axioms? So these axioms are a list of usual properties in dynamics which is uniform hyperbolecy, curvature bounds on invariant manifolds, distortion bounds, absolute continuity, and the growth of unstable manifolds. And then, I mean, this method was applied in many papers, especially in the case of Peter's with lots of success. And what we are going to do is kind of use the same scheme with a little tweak. And why we are going to tweak it, it's because in my list of properties here, there is this kind of uniform curvature bounds, which is kind of annoying because when you talk about curvature, you need objects to be situational. In particular, the invariant manifolds, if I'm going to talk about curvature, they need to be situational. And the point is that they are not situational in our setting. So this was noticed by Bauman, Brie and Burns in 87. So in particular for the profile that I wrote, 1 plus x to the power 4, glued with something that could prove that they are not situational at the closed geodesic. So you say, okay, you don't have the axioms for of Chernoff. But fortunately, in 99, Gerber noticed that it's not situ, but it's uniformity one plus leap sheets. And this is the right regularity to make the Chernoff scheme work. Because what he asked when he was talking about uniform curvature bounds, basically what he needs is a slow variation of the touch of its planes in order to compute measures of sets coming from projections of the tangent space to the unstable manifolds. And to that, you don't need the curvature bound, but you just need to know a notion of upper bound of the curvature. So in other words, that the tangent planes are not far into fast. And this is given by leap sheets. So in other words, even though we can't strictly apply Chernoff axioms as written in the literature, it's not it's not difficult to tweak them to include this slide generalization. And this is precisely the condition that we have for our examples. So we are saved by this work somehow of Gerber and Wilkinson. And now let me talk about, yeah, so this is the comment that was made, the curvature bounds is just to control variation of tangent spaces. And since he's just talking about upper bounds, leap sheet suffices. Okay, so now let me talk about the axioms themselves and how to check them quickly. So I yeah, the remaining time, I don't want to waste your time on lots of details, but let me just check a few properties. So the idea is the following, you are going to believe me that I just need to construct my section and understand my maps close to the neck because outside the neck, the genetic is just negative curvature. So everything is fine. I just need to understand how transitions on the neck happen. And to do so, let me just take the example of crossing geodesics. So I'm not going to try to treat the bouncing ones, but the three minutes analogous, let me just look at angles where you're crossing and coming out of on the side. And so my section, I mean, morally, but this is not true, but morally, the section that I'm going to take is just the following. I'm going to take these two curves here alpha and beta, then I can take some disks, SEO disks here on this part. And this will be my map. Okay, so this is basically the cross section. I mean, it's not true. Technically, it's hard to do that, but morally, what I'm what I'm trying to tell you is that to build a map to check channel flexioms, what I'm going to do is I'm going to take to take the vectors based on alpha union beta. And so every time across, I have a map and some disks outside just to complete the picture of the section. So this will be my map. But I'm going to analyze only the map from alpha to beta say, just to give a feeling of what kind of estimate to have to do. Okay, so we are trying to analyze what happens from up to angle. And so I'm going to introduce some notation. So as I told you, I'm going to say from the left to the right, so minus one to one. And then I'm parametrizing this geodesic by some interval of time. So minus CT so that at time zero, I'm really crossing the geodesic in the middle. Okay, so this is the meaning of these conditions. And now I'm going to look at geodesics whose collateral constant belongs to interval, which is pretty close to one. And it's given by a one plus I mean one over n plus one square and the one over n square. Okay, so I'm going to basically divide the set of collateral constants close to one into these intervals. And this is the composition. If you look at this, the way I'm doing this composition is very similar to billiards. And this is not a coincidence. We are really going to mimic what people do for billiards with the notion of what they call homogeneity bands. The idea is that we're trying to get uniform properties like uniform hyperboleity, uniform distortion, uniform growth. And there is no hope to get that globally on the face space. We have to concentrate on bands where guys kind of travel close together. And they feel the same hyperboleity which gets degenerate when they approach collateral constant one. Okay, so that's the idea. I'm not going to compare vectors which are too far. I'm just comparing guys in the same homogeneity band into this thing here. And so the first limit is that guys in the homogeneity band, they take almost the same time about the same time to cross the neck. And the time I mean almost the same time means a time of order n to the power r minus 2 over r where this approximation here is up to two multiplicative constants. But the exponent's correct. There is no loss in the exponents. Okay. And this limit is a calculation with clever row.  So this allows you to basically talk about a problem on our first order differential equation. We integrate it to get out the time and you check that to get the correct expression. Okay, so it's a calculus exercise. But of course you see, I mean, this is intuitive. I mean, the closer you get to the syntotic, the more time you travel because it's higher more and more for crossing and then getting back. Now, what is in the bag measure of the set of vectors in a given homogeneity band? So the measure is basically the set of this interval, which is 1 over n to the power 3. Okay. And why am I talking about that? Because this is kind of giving a control of tails of my tower. And so if you use these two facts, you can see that the time, I mean, the vectors is spending a long time in the neck. It's about this series here because you should sum over all homogeneity bands for which this number is bigger than your k to the power something. And this number is k to the power 2. So if you change variables, the time, if you spend time n in the neck, then the measure of those vectors is this 1 over n to the power r plus 2 over r minus 2 plus 1. Okay. And here you see the number that I was mentioning in my conclusion of my theorem. And so for the value, there's a kind of that number in the correlations plus 1, which is normal. For the map, you get always a number plus 1. We saw that in this lecture. For the map, you get a number plus 1. And for the flow, we are going one less. And now we need to understand the map. So the map here, you can explicitly calculate because of the symmetries of the situation. And so basically, we know how much you turn around in the theta variable along the geodesic.  And then the angle that you get in is the same that you get out by symmetry of the situation because you end up, you wrap around, you cross the geodesic and then you can reverse the mechanical system. So you get out to the same angle. So the angle coordinate is not very difficult. It's just the theta coordinate that gets wrapped around. So on each band, you are basically twisting the angle by a quantity, which it depends on the angle of entrance. And the amount of twist is a function that you can control very well. And this is crucial. This is why we stick to rotation surfaces, surface of revolution, because we can really compute this derivative. And there is no error in the error term in the power. It's really this guy up to two multiplicative constants. And for the second derivative, we just need that number bound. But for the derivative, we need a very good bound. And this is why we stick it first to examples. But an important thing is that since you are, so basically, this feature is for you approach the singular set and you twist more and more around this guy. So basically, you are kind of looking at the map of a surface, which is basically made of two components, a hyperbolic map, which is transitions in the negatively covered surface. And then a kind of parabolic map, a then twist, which is strength is bigger and bigger as you approach the singular set. And you are composing these maps, not randomly, but kind of composing them and trying to build a dynamical system, which is iterated function system of these two things, hyperbolic map, infinite then twist. So what is it that is getting bigger and bigger as you approach the singularity? This function zeta, which is a twist that you do in the angular coordinate until you go out out. I mean, the number of turns that you are giving here in this picture. And this this explodes as what happens as you get, oh, as the direction becomes more exactly, it explodes and we can control in a very precise way how it explodes. And this is important for our estimates and to ensure hyperbole. And this is why you are sticking to examples so far. I mean, for the moment. Yeah, but this is the idea. And in particular, we can compute, for instance, how unstable vectors grow.  So in particular, this gives a very nice uniform hyperbolecy. I mean, you expand at least by some amount. And this amount increases as you approach the singular set, the derivative explodes as you approach, which is good for expansion, I mean, and contraction. So you have this uniform hyperbolecy property. Then you can also prove using similar bounds. Since you have bounds on the derivative and second derivative, you can prove distortion bounds. And then you can prove in particular, yeah, that the log of the derivative, it's varying like a holder function of the points. And this is because we can control the derivative. So the quotient of z to primes by z prime. Okay. And so this one third here comes from the fact that the quotient is really n square, if you divide here. And the interval where this happens is the homogeneity band is n to the power minus three. And so the distortion is n squared by this.  So if you give up two thirds in the exponent, it remains a one third. And then you can also prove absolute uniform absolute continuity by the same means. Okay. And again, I mean, we are using a holder continuity of the foldation, which is not automatic. I mean, it's automatic in hyperbolic dynamics, but here we are again using Jabra Wilkinson. Okay. They have two theorems, one's giving lip sheets proper along the leaves and holder continuity if you move in any direction. And then as I told you, so these are let's say two axioms or three axioms, I mean, uniform hyperbolicity. Yeah, I have not explained for you the growth lemma, but I said about uniform distortion and uniform absolute continuity. But yeah, you have more axioms to check in particular, have to check that for instance, when you have the singular set corresponding to plural constant equals to one. And then you need to check complexity growth of the singular set. So when you iterate and iterate the set, the weight crosses and the composes the face space is not creating too many pieces and so forth. But yeah, but I think I'm not going to discuss it because this discussion is technical and especially because this discussion depends on a appropriate choice of the section which I did not do today. I mean, I was doing the calculation with this fake section, which is just the next. But actually, in reality, you have to adapt to that. The compose more and yeah, it's more technical. So this is why I'm not going to discuss the other part, because I'm not going to give the correct expression of the map. But you can kind of believe that out of these calculations we can get enough control on hyperbolic to get a map, a surface map with a young tower of exponential towers. And now you need to pass this information to the flow. But to pass the information to the flow is the computation that we already did, because I told you that transition time, so the time to cross is this, I mean, the tail is this, the set of the measure of guys which take time and to cross the neck. It's about one over into this concept plus one. And so if you convert that into the flow, you should take this one out, and then you get the correct, the correct key of correlations. Well, it's not so easy. You have to apply, I mean, if you want to do it properly, you have to apply recent work from 2019 by Ballant Butterlain Melbourne. But let's say morally, that's what it is. And I think I'll stop here. So I'm taking your questions. So thank you. Thank you. Thank you. So I try to say thank you in the language of the organizers. And so let's thank Matheus. Thank you Matheus. Thank you. Do you have questions or comments? We'll have one more question, but I've been asking so many questions. It's really, I mean, it's really just about this last part about going to the flow. I've not, you know, I've worked very little with flows. I did some work with Ian on the law and equations, but I remember that the problem with the decay of correlations is that you need, I mean, I remember that it's, you need actually to show that the flow stretches somehow, right? It's not just, it's not just an automatic issue. Exactly.  I mean, when you do the, so let me show some correlation. Yes. So like this one, when you do the Laplace transform, what you see is a kind of exponential i times a temporal function. And then you want this function to oscillate a lot to be able to produce constellations. And in our case, and this is what I mean, you can ensure, for instance, you should ask for the condition called uni, uniform non-integrability of defoliations, meaning if you do a path along the stable of epsilon size, then unstable, then stable. And then what, when, and then unstable, when you get back to the flow direction, you get by distance of epsilon square, say. So the defoliations are not, not only non-integrable, but they are non-integrable in a quantitative way. Yeah. So, and this kind of quantitative non-integrability allows you to run this stretch of things and pass to the flow. But in our case, my claim is that this is automatic because we have a context structure. This argument that goes back to, I mean, this kind of application of context structures to get statistical properties go back at least to a paper of Katalk. And it was used and reused by many persons. I mean, I'm not, I don't know if I'm historically accurate in saying, but I remember clearly a paper by Katalk doing that to get Bernoulli property. Yeah. And so here's the same, I mean, here you have to take advantage that we don't have to discuss much because our flow is already geodesic, so it's context. So it suffices to know the tail. Yeah. But in general, you are absolutely right. We've had, if we had abstract flow, we had also more work to do to check side of conditions, non-integrability conditions. Okay. Thank you. I wanted to make a comment that was not in the slides that the paper of passing from 77, it proves actually the Bernoulli property. Oh, that's right. Yeah. Yeah. I started talking about the DC and yeah, absolutely. Yeah. That's right. Yeah.  There is another thing that I wanted to point out. This kind of infinite bent twist map that Matteo said is very hard to work with because you have a singular set and if you have, if you get a curve that is transverse to the singular set, the image of this curve of, which has finite length will actually be infinite because its image, it will rotate infinitely often and accumulate in a closed curve. So it makes it very complicated. The analysis, even for this complexity bounds, because you have to have like no triple intersections, but you are dealing with curves of infinite length and you don't know what happens when you perturb the boundary, what happens to this, the new images. So you have the perturbative argument that we make, it's not as standard as we thought and they actually make many mistakes to get to the right way of constructing the section. Yeah. That's right. I mean, so this is why I said that it's what I showed is that kind of fake cross-section because the true cross-section has to choose well the boundaries to ensure these transversality properties that are known as standard and relies on the formula for the map really. So this is why we are sticking to the example, because at least in the example we control everything. So I mean, of course we believe that this could be generalized, but we are not yet there. Do we have any other questions or comments? Okay, so let's thank Matheus again.


After 15 seasons of success, the one and only U-Pike women's bowling coach Ron Dameron has stepped away from the lanes. The bowling program was started in 2000 and I've been here 10 years when we started the bowling program. And at the time I was athletic director and also dean of students here with that. We were just looking for another way to get people on campus and looking at sports. And we ended up starting bowling along with football and women's golf together. We got it kind of started, kind of started on the slow win. And now it's a conference sport in the Mid-South Conference. We have 14 teams next year that has bowling in our conference. It's becoming an NAA sport now. It's getting there. The NCAA has made it a championship sport now. So it's really, it was a fast growing sport and it kind of got to it. And we just got into it at a good time. In his time with the U-Pike women's bowling team, he has led the Bears to six national championship appearances, four national titles, two NAAI championships and was named USBC Coach of the Year four times. Some things happened for a reason I guess and I just kind of stuck around. And I had no clue what to expect. I really had no clue that we would ever win a tournament. Sure, I didn't think we'd hardly ever make a national championship. And I'll tell you, I have been blessed with such great young people in our program. And I think sometimes God just puts you in a place that you need to be sometimes. And I think that was here at the university. And sometimes if you do what you're supposed to do, then good things happen. And I think we've just totally been blessed with how we see things work here at our university. A search is currently in progress for a new women's bowling coach. But the shoes will be tough to fill after the success of the Ron Dameron era. Reporting in Pikeville for EKB News, I'm Makayla Colley.


Hey, how's it going everybody? My name is John Hammond and I'm looking again at Python challenge A lot of people have asked me to get back into this so I figured I would try my best and here we are looking at challenge 16 We just finished looking at Mozart stuff now. We're granted with this image and honestly I have no idea what I'm looking at right now this looks like just a bunch of pixel colors and Kind of scribble stuff looks like a bunch of static Although I do see a lot of these like weird pink purple things like all over the place Looking at this initially I want to you know, I'll view the image so we can kind of zoom in on a little more Move it around. Okay. I guess I can't really do that that well Let's grab it and play with it in paint or something I'm putting all these in a specific folder for Python challenge You might want to do the same thing. I'll just put it there and then I'll open up color paints to actually open that file And I'll zoom in on it a bunch Okay So, I mean I can see like lighter pixels like we can see those lighter areas that just looks like static and It looks like these like pink purple dividers are on every row, right? Like we can see them on literally every single line in the image But these color values are so distinct. They probably aren't going to translate to like ASCII characters I don't feel like Since there's only one of these pink dividers on like every row it looks like At least from what I see are these on these are pretty close. So these on two different rows two three four and Two three five. Yeah, they're all on different rows. There's only one of them on each row I don't feel like it's like a divider between a bunch of these random color characters that could represent like ASCII texts to generate a message I don't feel like that's it. I Don't know what It could be referring to there's nothing.  They must go somewhere together I wonder if all of these like pink divider spots have to be aligned and like put together in a Vertical sequence, you know, we can probably do that We can definitely do that in Python with like the image library and the image like operations library If you haven't seen that we can just like move These into a specific position with an offset and all the data like every single line will be put together for us We'll just have to like chop up and like crop each individual row. We could totally do that I guess let's try that just to poke it stuff just to see if there's anything we can really do See I'll discard that Am I still recording? Okay, I Want to make sure that I actually still have audio before I do this. Let's try that. I'm just gonna pause this Okay, we're good. I still have audio. I've done that before I've like started to record an entire video And I've been like I've put it like 20 minutes or 20 seconds 20 seconds now Like 10 minutes or 20 minutes into a whole like programming thing and I turned out wasn't even recording sound so All right, let's start like a level 16 a dot pi file And I'll try and amp this up so you can see it It's out of shebang line Because for some reason I can't get file header do that for me anymore and let's open up this image let's get image and And That's really all we need for now Image can equal image dot open and the file name is a Mozart dot gif right and let's close that image And let's get just get the dimensions of it because I'm sure we're gonna have to like actually loop through it I want to actually open it up in another window Don't look at my home directory. I got a bunch of weird stuff in there. Oh No, I'm the completely wrong thing Where did my image go? Did it just go away? Was it hidden? All right, now we have the image that we can just fire up. I also had in the web browser, which I guess I could have done Sorry guys. I'm off to a slow start today. Let's get the size. Let's get the like width and height out of this image and Since this is a gif right if I actually print out the image info you can see clearly This is a gif so it's gonna be a little bit harder to like process images and like process colors out of it We can't like loop through it and get the data that easily. So Let's try let's try to do that that that row separation thing I'm gonna do that for why in range of height So I'm just going like line by line here by rows and I'm gonna set up a box crop box These are just gonna have to be the dimensions because if I actually get the documentation up for Python image Crop that actually needs a box and the box has to be a tuple defining the left upper right and lower pixel coordinates So we'll start with zero being the left Y being the Upper right being the whole width of the image and then the lower pixel coordinate Will be y plus one because we're just gonna go just grabbing like a line for it So row can equal image dot crop Pass in our crop box And I'll just do this for the first line just so you can see what I'm doing actually Now I'll do row shows and that's gonna be an image and I'll run this and you totally can't see that So this is like the whole row the whole first row just expanded out a bunch I don't know how well you can see that. Whoa. That was a weird thing that I did I don't know if you can see that at all regardless You can actually get the like the text out of it. We're gonna do row Dot two string since this is not Like a PNG or anything. We can't easily get colors out of it So this will be okay Obviously a bunch of jazz that we can't see in sublime text until we make it representative of the whole string And now we just have a bunch of like escape sequences that totally represent the data Now we just have to figure out if we're trying to find that Position. Oh jeez in the image where we actually have that like Pink and purple Indicator like that offset indicator. We're gonna want to be able to grab that position, right? So let's do that We have to find what it actually is though first. So since we have this in our String variable. It's actually fine and we can work with it like a string We just got the index like a string if we just treat it as a string. So let's say Row string can equal obviously the row two string and now for I'm just gonna loop through this so we can get like sequential characters It's just kind of like a quick Cheap hacky way to find out what the character might be so for I in range of Roast length of row string right because we're looping through it and I just want to be able to use it I want to be able to have I as a number not like for character in row string I'm gonna use I so I can index the sequential like the characters that follow This is gonna get me an error But it's just for like a little hack to find out what character is repeated because obviously we know that That offset indicator that like color that purple color is gonna be repeated Since it's a little bit wider than all the other things so character Equals string I Row string I sorry and if Character is equal to Row string I Plus one so we know it's the same thing as a sequential character and plus two and Plus three so we're just like looking forward to see hey Is this character repeated if it is what we can do is we'll just print out the representation of that character Just so we get the raw bite of it And it's gonna give me an error once I run this I know that for flak But it's still gonna give us it's still gonna find the pixel for us right yeah Yeah, so we get an error as expected, but we see that xc3 is repeated That's the one that's being repeated And we get the error because we're probably just gonna get out of range after once we hit the end of the row That's fine. We now know that this like pixel right here that character When you look at that pixel as all characters that is our offset indicator now that we've grabbed that we should be able to just find Where that offset is right? so I'm gonna remove this row dot show line and this finds us our offset indicator We'll comment that out because we don't need it anymore But now we can find our offset to equal row underscore string Thought index because we want to find where that actually happens that's offset indicator That's gonna be the purple thing right? But it looks like these all start at a white character that happens right before it so I'm just actually gonna Subtract one from that offset indicator Just so we know that's where it is Now we can actually take this offset and wrap around our whole row with it And I can do that with the image ops image channel ops are in Python image chops module offset Is what we want to be looking at? Try and find it for us So offset will return a copy of the image where data has been offset by the given distances So since we just found our offset indicator and where that offset is What we can do is we can just shift it over to like where the position of that Pink thing is so it will essentially align them for us wherever we want them to the data will wrap around the edges And it says if the y offset is emitted and sort seem to be equal to x offset That's what we're gonna do since we're just gonna pass in our original offset and Have it assume that hey, we want to remove x we'll do this for our row. We can just say row equals I Gotta actually import that module Import image chops And we can say row equals image chops dot offset What does it need it needs the image which would be row in this case and it needs the offset Which would be the offset that we just found cool So now we have that row shifted, right? But now we actually have to put that on our image that we're working with and we can do that since we've cropped the thing We do that real easily since we've cropped it With I think it's image paste Python image Till I just want to find the Documentation for it Paste is that is it? Yeah, yeah, yeah paste another image into this image the box argument box is what we just have because we cropped it, right? Yeah Box everything is a tuple or for tuple so we can just pass in that crop box that we just sent So we'll know where to actually put it so we do image dot paste our current row that we're looking at and where it has Been cropped to you with crop box so Now that we're done with just this single row we can image show this and I'll hopefully if I display it out. Oh No, it's not working. We're not getting it over there Do you guys see why that is? If you think about it this offset is actually like a positive number obviously So when we shift that by that offset, it's in pushing it forwards pushing it to the right but since the Pixel since our row indicator that purple line thing is already At the I want to find it. I want to find where that image is cutting it Gonna open it again code Python challenge 11 since It was already so far to the right. We have to move it to the left So we actually have to pass in our offset as a negative number so it'll subtract from its current position, right? Let's do that. Let's do row to offset now when I run this you can see Just barely I can't really zoom in on this like oh god image magic does not like control plus Just barely hopefully you can see that it has been shifted to the right all the way to the left Sorry all the way to the left so it's at the front position in our image. Let's do that For every single Row, I'm just doing it right now, which is the first one But if we were to do that I wonder since it'll align all of them. I wonder what will give us I'm not gonna do image.show for each iteration. I'm gonna bring it at that at the very end And let's see control beat Whoa, there it is Romance. Oh, that's totally it romance is totally the next little flag or whatever we can use to get to the next level That's cool. Look at that. It just ran through every single row and a line just at the beginning where That like indicator is that that purple box thing Try and display it again Can I zoom out here? Yeah, look at this every single purple thing was just moved to the left corner We've aligned them all and we get romance as our as our output With all those like lighter color lines being put in the right position. That's cool. I Want to I want to play with that a little more honestly I want to make a jiff out of like because what it's doing is it's doing this line by line, right? Let's I'm totally making a jiff of this. Let's do that import sub process because I'm gonna call image magic I'm gonna have image magic. Just take all frames of this. Let's actually take a frame number And Let's do frame plus equals one Just so we're adding a frame and then I'll save the current image as Frame zero plus the string the current frame we're on and let's just make it a dot jiff How do I and then we could just use Python to like make that into a jiff Python jiff programmatically this thing this thing I Just saw this a little bit earlier some guy just like took an image magic line So he's calling it from the command line And we can use print sub process. This probably isn't gonna give us any output since it's just calling a Check outputs. It's just calling a check up a thing And I want to have no delay and I want the frames to be frames anything dot jiff Animated jiff and some sub process wants this as an array let's just split it by each space character and And then we should totally have a jiff at the very end Sweet, let's try and run this. It's probably gonna take a little bit. Oh sub string not found. What is that? What went wrong? What went wrong? Okay, after a little more testing I did a little a little bit of thinking and for some reason it breaks when I'm saving it as a jiff, so I'm just gonna save it as a PNG and Change it to little takes PNG files as well Also, just gonna take that string and Z fillets just in case We have a little bit more than zero to it just so it's all in the proper order So I don't we don't need that trailing zero All right, so now hopefully we can run this it's gonna take a little bit It's gonna take a long little bit Because it's just savoring every individual frame and then creating an animated jiff with it But once we see our picture show up since we get the final romance thing hopefully it will look good. Oh God dang.  What's going on? Frames, oh Is it because I removed that zero? Frames Dang it. I forgot an S. Control B run it again. I'm off my game today guys. I'm sorry If I don't challenge is hard though, so I've been taking me so long to record this stuff because like I gotta figure out What's going on too? Hopefully no errors this time No errors. Yeah, cool. All right, we get our image now If I drag over my folder, it's literally filled with all these frames if I open up the animated version It's gonna take a little bit to load because there's so much stuff to it. I'll make it full screen and Oh, look at it. Look at it. Just pile those frames on that's pretty cool. Just unravels That's how you seduce the computer right there Unscramble bits get romance to appear Cool All right, thanks for watching guys. That's got us our new Python challenge thing. So let's hop over to your Python challenge calm for this PC for such a turn Romance dot HTML and here we are. What is this? All right, that's what's next everyone Cookies and weird stuff, but thanks for watching. Hope you guys enjoy this one I know it's been a while since I've done some of Python challenge, but We're moving along See you soon everybody


seems like we have a different definition of freedom with David, because David's definition of freedom is he tells Saul I'm gonna go and show myself in front of a Goliath. Goliath is not bothering you. Is he tormenting your family? David? Has he thrown a stone in your face? Has he punched you? Has he hurt you David? No, he hasn't done anything.  And David says it's simple. How can God deliver me if I don't pick a fight? He says I will go pick a fight and God will deliver me. Don't expect a deliverance if you refuse to fight.  We cannot expect God to move his throne if we can't move our pinky. And we have this attitude, this passive attitude in Christians today. What is this? You know if God wants it, God will do it. Apostle Paul tells Timothy, fight a fight according to the prophecies given to you. I read that as a Paul. God said it, God settles it. If God prophesied, don't do anything. It's gonna get done.  Well Apostle Paul didn't hear that preacher because Apostle Paul tells Timothy if God said it, nothing's gonna happen until you move, until you fight with it. God gave all the promised land to Israel but they only God as much as they fought, not as much as God gave them. Like we heard a testimony today, God gave all the blessings on the cross but you only get not what you wish to have, what you fight to have.  But if you're waiting for every prophetic word, for every seven green lights to be on your way to work, for the proper verse, for the proper sermon and the proper song, for everything to line up and then you have the courage to step against the Goliath. My friend that day might never come. You have to launch and attack and see God will deliver me if I throw myself in something bigger than I am. If somebody say amen, most of the healings Jesus did, they didn't happen in the hospital. There were hospitals. Why didn't Jesus go to the hospitals and heal sick people? People always say well if he is such a healer, if God used him to heal people, let him go to the hospitals and just heal everybody out of the hospitals. The Lord Jesus didn't do that. Who did Jesus heal? People who threw themselves when he was eating, interrupted his lunch, when he was walking and they broke every protocol that said you run clean. You can't touch clean people and they shattered that and they threw themselves in the places they could have get stoned for and that's when God met them. Very few cases where Jesus almost an accident stepped in. But most of the cases somebody did the fighting and Jesus came as a result of it. God is not gonna show up to your battle if you run from it. You can't send God to fight the Goliath if you're hiding behind the bushes and snapchatting. Gonna put the phone down, pick up a sword, go against the Goliath, you're scared to death, your heart is sinking, going into your feet and you say God if you don't show up, pooping in my pants is gonna be the least of my worries. I'm gonna die in this battle and that's when you really your faith is stretched.  Well you know I'm just I'm just really praying that God will just God will help me. If David would have listened to that he would have been embarrassed. Sometimes we have you know generational curses I just I came to prayer line once. That's great that you came to prayer and that is awesome. The question I have for you today are you throwing yourself in the battle bigger than you? Trusting God to deliver you. Or are you waiting for God to do the deliverance so you don't have to do the fighting? You're more than a slave. You carry a DNA of God. You carry an image and likeness of God. Lions make lions. Eagles make eagles. God made us in this image. Never lost the battle and he loves war. He is a man of I will say he trains my hands for battle. He trains my fingers for war. God creates you. He finds pleasure in seeing his child rise up from the diapers and take their position and take against the enemy. Jesus didn't just give us salvation. He gave us authority.


Hi everybody, good morning. You guys know that impact investing has been around for many decades, and it's grounded in a bunch of trends around community finance, in housing and small business in the US, micro finance and international development, and foundation programs and mission-related investing. We have been working to, oops, get this, do we have our slides? Green button. There we go. So some of you may have seen, last, earlier this month, there was a very interesting debate on the Stanford Social Innovation Review. Paul Brest and Kelly Bourne wrote a piece about, are we really having impact in impact investing? And there was a ground swell over response. There were 18 field leaders who responded and gave their honest commentary to what was going on. And we looked at this and we said, you know, this makes it look as if there's no cohesion in this marketplace. There's a lot of different opinions, and they're not consistent, and it's a little disturbing. The reason it's disturbing to us is because we've found out that this is simply not true, that in this marketplace we have been looking for the past two years in depth at 13 impact investing funds around the globe that work in 80 countries that manage over $3 billion, and we have hand-selected them because they have what we're calling exceptional returns, which means they have met or exceeded their investor's financial expectations, and they have met or exceeded their social impact expectations as well. And what's exciting is that when you look at, across these funds, what we're here to tell you is there's a lot more cohesion, and there are practices that go across all the funds, and it seems as if there are some signals for success. That's what our project has been about. It's called the Impact Investor Project. We are going to be releasing a report later this fall on the project, but what we want to do this morning is give you a little bit of a preview of some of the themes that have come out of this project. We're going to actually be talking later this afternoon from 2.45 to 5.00 in the fleet room, some of you have seen these little cards on your chairs. Some of the fund managers that we've been talking with are going to be there this afternoon, and we hope you'll join us to meet them and hear their stories directly, but in the next 10 minutes we're going to talk about what are some of the practices of these real funds and debunk some of the myths that we think have been holding the field back. We're calling this Impact Investing 2.0. We feel like 1.0 has been this mishmash of opinion, and it turns out that there are common practices that we think will define Impact Investing going forward. So I'm going to let Ben take over with our first... Thanks, Cathy. So this is a fund, an investor, sailing towards the financial and social objectives it cares about, and it's both an inspiring image, but it's also a little worrying, inspiring because it's unconstrained. This is a platform for innovation. The boat can set its own course, but worrying because there's no support for when the boat hits more challenging times, but this is the mythology, that this is the ideal environment in which to innovate, and that unconstrained innovation is what is most desirable. What we discovered when we looked at the funds is that this is a mythology. The truth is that the boat is not alone, and this image hints at all the institutional supports that really underpin Impact Investing. But the one we have in mind is the role of government. Three of the 13 funds we researched were created in partnership with government. Eight of the funds we researched had investments directly from public sector institutions. Six of the funds are structured and have capital heavily influenced by government regulation, and seven of the funds work explicitly to influence the policy environment in which their investors operate. So in short, Impact Investing is in fact a form of public-private partnership. The public sector is ubiquitous. It's everywhere in Impact Investing. Newer generations of funds, more mature markets, or particularly underdeveloped markets, may exhibit fewer characteristics of government involvement, but the reality is that government, as you'll hear later this morning, is becoming more interested in Impact Investing, not less interested. So Impact Investing is public-private partnership. Cathy? So the next thing we want to talk about is the role of catalytic capital. So often when you look at how Impact Investing deals and funds come together, it looks a little bit like this jumble over here. And in fact, Audrey Choi of Morgan Stanley has called it the crazy straw picture of how different kinds of capital gets together. And a lot of people look at this and they get really nervous. There are some people in the philanthropic community that say, you know, unless it's very clear that we are needed, we shouldn't be participating in this market. There are people in the mainstream market saying, unless this is a clean, mainstream, straightforward investment, we shouldn't be in here at all. And what we have found in these, again, 13 successful funds is that this notion of catalytic capital is essential and has been essential to them succeeding. So the picture is more like this, where that interweaving of different kinds of capital with different kinds of goals has led to the important impacts that these funds have been achieving at scale. And there are a bunch of different ways that that catalytic capital comes together, and we're going to be teasing those out in our reports. The important thing is that it has different functions. Every single fund that we have profiled has catalytic capital in, and many of them have catalytic capital going out, where the capital is an instrument for generating returns, and it works to either sustain, de-risk, signal, seed, or align with other strategic objectives. We're going to turn it over to Jed for the next one. So I love coming to SoCAP, and the reason I love coming to SoCAP is that I get to see all of you. And all of you represent different parts of a really diverse community of actors that's coming together around this common challenge and opportunity of how do we drive capital for multiple returns, for financial performance, for social and environmental impacts. And one of the things that I like to see is finance people coming into the conversation for the first time. I like to see kind of reform social workers coming into the conversation. I like to see the mix, because I think that's really the strength of this community. And I think that what we've seen, though, in the past years has been a devolution of the conversation and a myth that seems to be coming up, and I think in some ways it's tied to this, are you a finance first or an impact first conversation? But it's a conversation that says the parts are more significant than the whole, and that what we're really trying to do is convince somehow the finance people to do more good, and the good people to be more financially responsible. And we are at risk, I think, of continuing a division of silos instead of cross cuts. And one of the things that we have found in our research with these funds has been that the most successful funds have actually been led by fund managers and certainly by teams that actually are multi-wingual. They're able to comfortably play across those different silos of government orientation, private market orientation, and development orientation, in a way that really allows them to see a new hole, to see new opportunities, to create impact and manage their capital for impact. And so the successful firms are really those that think in this integrated way instead of the parts. And that's one of the pieces that we're going to be exploring in our session this afternoon is what we're calling the multi-lingual leadership, or what I used to call mutant managers. Because these are folks who basically don't fit in either side traditionally, but fit very well in that cross-cut space of advancing blended value through capital allocation. Thanks, Jed. And finally, this kind of abstract image of a couple of lines. This is the niggling, constraining theoretical underpinnings that I think a lot of us have become more conscious of, which is what Jed described as the dichotomy between a fund being a financial first fund and being an impact first fund, which really defined the era of impact investing 1.0 and was very useful in enabling an initial conversation. But we believe it's time to retire that dichotomy. Financial firsts, you know, were funds that were set up to achieve sort of uncompromised sort of market rate return impact first funds having an impact intention which was disqualifying in terms of an investment and an interest in the investment. What we've seen with all 13 funds as the case with catalytic capital, where all 13 funds had catalytic capital at their core, all 13 funds approach impact investing in the same way, regardless of their financial or social proclivities. What that actually looks like is this. We're at the beginning of a fund's creation and the origins of a fund and investment thesis and investment theory of change is developed which embeds financial and social impact. This is essentially a mission first approach where we're using impact investing as a tool to accomplish a particular set of financial or social objectives. Once that is established and that is in a fund's DNA, the fund then moves forward with absolute financial discipline. They must move forward with absolute financial discipline. That's what the most successful funds do. And then during diligence, deployment, they have that financial discipline, the top of the line here. And then towards the end of the life of the fund or the end of a life of a deal, they return again to that initial investment thesis which was based on their underlying mission and objectives for doing impact investing. So in effect, impact investing is more complex. There are institutional objectives. There's institutional structures that govern how folks deploy capital. There are values. There are financial objectives. There are markets that we care about. And all those things come together in a much more complex and nuanced fashion. The shape of the arc with financial and impact considerations overlapping is the factual, practical underpinnings of the way funds do what they do. But essentially, that's representative of a much more complex way of approaching this. And we think we now have the tools at our disposal. We think we now have the data and evidence we need to move to a new era of 2.0 in impact investing where for the first time we can really understand that nuance. We can tackle that complexity. We can segment the market in a way where we have the knowledge to truly align markets with mission. So please join us on the journey towards impact investing 2.0 and we look forward to seeing you this afternoon. Thanks so much. Thank you. Thank you.


Alright, welcome to my new flat tour of 2023, here living in London, me and Nora the person behind the camera have been living in the same house for the last five and a half, six years or so, so we're very very happy with this change, it's finally gone into the point where we've got all the furniture in place, I think it looks lovely and I'm really excited to show you around, so let's go! Come on in, immediately a much bigger entrance than you probably remember from the old place, if you've been watching the videos for some time, right over here in the entrance hallway, we've just got some storage closets over here, right now we've got a lot of random kind of knickknacks and things that we don't want out and about in the house, boxes, a drying rack for laundry and all kinds of stuff, probably not that interesting, so we'll close it right away, on the right here we've got some paintings drawn by our mother, this is our dog Kitsune, dressed up as a man and this is our cat Simpson back home dressed as a lady, which we think are really nice, so keep them in the hallway, and then over here this is kind of a coat closet, but it's become my coat closet because I don't have enough space in my own room, there's also some storage for a few suitcases and other kind of things that we want hidden and out of the way, so this is kind of the entrance area over here, on the left we've got, what I think looks like a really cool kind of open design shoe rack, we've got mine on the left hand side and then Nor's on the right and it's almost a tall as me, which I think is kind of cool, we've got this bench over here for when we're putting on our shoes and things in the morning and of course how can I forget this glorious glorious floor mirror, this is something that I've wanted for a very long time and so as soon as we moved here and we had a little bit more space it was at the top of my list of things that I wanted to get, so very very happy with it, and then the first room over here, this is my bedroom, I don't think there's too much interesting to show in here, pretty small, there isn't a lot, a lot going on, it's basically just a bed, closets, a bedside table, you guys might have seen from the video where I moved in and unpacked my house a little bit of my closet, but it kind of looks like this, neatly organized or I like to think so anyway as much as I can and there's a good amount of closet space but just not enough for all of my jackets and things, but anyways there's more interesting things to see so let's move out, hopefully you've noticed all the nice greenery in the house so far, we've got this like coconut type plant over here and we've got quite a few plants in the house, Noor takes care of most of them to be completely honest, she does a lot of the watering and caretaking and debugging and nutrition and all of that but I just make them look pretty and put them in the right place, we've got this snake plant here and then this this console I really really like, so over here this is kind of like a catchall I think it's called, so when me and Noor come home we just empty our pockets, we put our like wallet, car keys, earphones, badges for for our different workplaces etc here and that way we never lose them, when we come to leave the house everything is always here, it's easy to pick up and we can go, all right so moving on further down into the hallway we have the guest bathroom and my bathroom over here, a couple of shelves on the side, my office and then Noor's room bedroom and office, so I guess we'll start over here, we've kind of just been keeping a lot of the books that we have in the house, this is mostly my medicine related stuff, a couple of suture kits and that to me cards and things and then this is Noor's all of Noor's books and notepads and things like that, this is a poker set which I'd love to use on the lovely dining table that we have inside which I'll show you in a little bit but coming around this way you guys will probably recognize this artwork from the old house from my bedroom slash office slash studio slash chill area everything and then this is the bathroom slash guest bathroom might be a little bit loud because the ventilation fan is on and stays on for quite some time it's not too much to see here all right just all of my toiletries and things over here hopefully neatly organized as they should be then over here all of our important files and folders this is how I know I'm almost 13 okay I've got a whole array of colorful files and folders you guys will remember a lot of these from my exam notes and revision notes throughout my years at university but then now we have things like banking and taxes and work related files so feel like I'm getting really old not going to lie at least Noor takes up half of this shelf so behind the store over here we have my office this is where I would like to say that I do most of my work and spend most of my time but truth be told I just like taking my laptop and sitting on the couch in the living room or on the dining room table and I just feel a little bit more open and comfortable over there I've put up a bunch of these posters and plaques on my wall we've got the silver youtube subscriber plaque and we've got the gold youtube subscriber plaque these are posters that I had from my time at the University of Toronto this was a gift from Noor actually from Cambridge from a little while ago that's my medical school diploma this is my GMC full registration my license to practice medicine here in the UK this this is funny this is mishmish's favorite blanket so I don't know if you've seen mishmish in the background maybe some of the video clips but this is where she likes to sit as you can tell by all the unhealthy amount of hair that's sitting on there and then this is my desk it looks very similar to what you guys will have seen from previous kind of desk setup videos and I go into a lot more detail about it in those videos so if you're really interested in this kind of stuff and you should check that video out it's a standing sitting desk which is great for when I need to study and spend long times sitting down it's good to get a little bit of difference between that sitting and standing and you know massive screen great speakers cameras I love it let's move on shall we do your room yeah I'm going over here we come into Noor's room which is the master bedroom of the flat which is such a nice person I just had to give it to her so if you remember Noor's room from the old house the layout is extremely similar I think if you were standing in that position in in the old house looking at Noor's room you would see pretty much the exact same thing it's just a little bit bigger in this dimension it's a much nicer view and it's just a little bit brighter I think and then she's got her piano here where she's currently learning the what are you learning to play succession theme the succession theme song yeah which is really really cool Noor has a wonderful array of very large closets so it doesn't need any outdoor space uh like I do and then Noor's bathroom I don't know do you want to yeah sure give a little pan go ahead give them the tour you tell them behind that all right guys my bathroom much tidier than Nassar's got shower all the things you need I'm doing an excellent job as camera woman thank you very much there's that good job you smashed it all right and now the best part of the house our favorite part where I spent the most time the living room let me introduce you to everything going on here I'm I'm so happy with how this living room has turned out I really love it I think it looks very aesthetic in my own personal opinion I feel so relaxed and comfortable when I lay down and I sit here after a long shift at work just something I never really felt in the old apartment so it makes me very very happy to be here and me and Noor talk all the time about how grateful we are and how much we we love this apartment so I'd love to show you around I guess over here on the left we can start with the dining room table we went for a circular one as opposed to a long rectangular one mostly because I didn't want a huge object over here blocking all of this open space and I really like it I'm happy with it it's this beautiful wooden top which means I am very anal about everyone using coasters so all my friends who come hang out here they know very much about how often I'm telling them to put things on coasters and this is kind of my chill spot so this is why I do most of my work I mean my laptop is here this is why I sit down and I do my work I've got my pen and paper my sticky notes and things and I just have this big open view I can look around into the living room and feel I don't know more relaxed than I would if I was sitting in the living in the in the office this here is one of my favorite pieces of furniture it's like a what are these called it's a floor lamp with like an arc that goes the arcs I guess on over the dining table I think it looks really really cool and it's also something I wanted for a long time kind of like the floor mirror outside and then if we pan around this way we've got plants galore it's a joint effort between me and Noor I won't lie Noor does most of the caretaking of the plants so big shout out to her and these are all supposed to be non-toxic for cats because Mish Mish likes to chew on all the different plants that we have here but so far everything has been okay clutch wood over here we've got a little board games area for when friends come over and things just see them easily pick them up and play whatever we want nice big TV a couple more plants floating down on the side this is our favorite one what's this one cold one I think string of pearls string of pearls yeah this is a big favorite p.s.5 obviously and then these are the couches the reason this video has taken so long for me to make and put out is because this couch wasn't here I wanted this couch to be here before I made this video it's a huge piece of furniture that I was really looking forward to and really waiting for and I didn't want to make this house tour without it because it didn't feel complete to me we've got this lovely coffee table over here which matches the dining table and then this is the couch that we had in the old apartment the one that folds out into a double bed so when our parents come to visit us when we have friends visiting from out of town they can always sleep here comes out to very comfortable double bed honestly more comfortable than my own so let's come around this way over here by looking out oh I almost wasn't going to talk about the carmetic sign I almost forgot about it so this is the carmetic sign from the old apartment I didn't really find anywhere else to put it in the house and I wish it kind of fit flush over here and that I could plug in the cables I could turn it on but unfortunately none of those things are or the case but we just have it over here kind of sitting sideways and it's a really nice reminder of the old apartment carmetic channel how far we've come etc I don't know what do you think about it I like it I don't mind it you don't mind it okay it's a pretty it's a pretty big object in the living room yeah it's a shame it doesn't fit perfectly um because it's so close coming over here we've got more plants okay look we're taking care of them they're alive they're healthy okay we're doing well responsible thoughts there's an air purifier and humidifier here because believe it or not I am very allergic to cats and dogs and when we got mish-mish our cat sleeping over there I magically was not allergic to her when we went to visit the home from where we got her and I held her and I touched her and I rubbed my face I didn't sneeze I didn't start crying I didn't have a runny nose which is a miracle it's an the first animal I've ever interacted with that I haven't been allergic to so she's an absolute miracle but we'll go say hi to her in a bit we've got a printer here and then one thing I want to show you is this robot vacuum all right this is from a company called Roborock this is not a sponsorship this is not a blah blah blah this is just probably the best purchase I've ever made in my life after my cat obviously cat is cat is the first purchase but this robot vacuum literally just cleans the whole house by itself every day at midday and we come home to a clean house there's no dust there's no like crumbs there's no kitty litter from mish-mish nothing it just automatically cleans the house and it's like the best invention in the world just really really love it we've got a fan over here and then some of my workout equipment from when I'm working out at home but hopefully I'm in the gym most of the time and this over here is kind of the mish-mish corner so our cat is called mish-mish over here we've got infinite resupplies of the sticky rolls because she drops a ton of hair we've got all of her toys in here like combs a nail clipping stuff all that good stuff and then down here so this is mish-mish's litter tray which produces zero smell one of the biggest concerns we had when we got a cat in an apartment where it was going to be an indoor cat all the time was you know would everything smell and it has been completely odor-free right like yeah like literally nothing speak of the devil oh got you so this is mish-mish okay so mish-mish means apricot in arabic squirmy and we named her that because of her eyes didn't we she's very very not wanted to be cuddled okay okay okay you can as I was saying we're very very happy with the with the litter situation we just got a little bit of grass here so that she can feel like she's outdoor I guess and kind of need to find this area as the mish-mish area my god there's a lot of hair over here we've got mish-mish's scratch post which hopefully prevents her from scratching the couch even though that's not always the case and her little bed and then these are her fish all right so we've got like five or six little fish for her that I just laying around the house which she loves to play with sometimes and she kind of just moves them all over the place and they're her favorite toys and I just love the idea that she plays with little fish moving along we've got a little hammock over here for mish-mish so that she can lay down and stare out of the window and do her people watching all day and her little cat castles that these are called cat tree cat tree which I think fits the aesthetic of the house really well this is kind of like brown green with a gray and she loves to use it which makes us very happy because we didn't just buy it and she never uses it keep hitting this bird which is her other favorite toy and I think that brings us full circle if we turn around in here we've just got some more plants on the side a little water fountain for for mish-mish and then it comes back to the shelving area and the last thing we need to do is take a look at the kitchen so let's go there on the way to the kitchen we have the evil eye that my mum has very nicely placed or I placed but she bought and keeps all the evil spirits away and all the negative energy in the house hopefully keep us in in good health and happiness and all of that coming in here there's a pull-up bar that I bought the other day which I'm very excited about I don't know how excited nor is about it but it is what it is it is what it is all right very quick kitchen tour we've got like a laundry basket where we put most of our recycling like cardboard boxes and papers and things like that we've got this bin with a little wave to open thing which is a quality of life improvement that I didn't know that I needed until I had it little fruit basket everything here blah blah this is probably my favorite kitchen appliance I think the color is just beautiful it's like pastel green with the wood got our coffee machine that you probably remember from the old house and then this is my favorite this might be my favorite closet in the whole house actually wow so neat excellent smashing it I'm too short for this so we've got our plates and bowls up over here we've got pots and pans down here on these very cool things to kind of swing out for easy access hop toaster coffee machine all of our stuff washer dryer dishwasher sink blah blah blah it's just a kitchen in our fridge I don't think there's anything too interesting oh actually one thing I wanted to mention is dash water still needs to sponsor me I posted a picture of you on instagram like a couple of years ago saying sponsor me and you know we still have bigger things out so if you're watching tell your people to reach out to my people I think that's it that is pretty much the whole house I might have rambled a little bit through it so bear with me I'll include b-roll of everything that we kind of talked about over here give you a little bit of a better idea of how the space is used but yeah we've been here for a few months now and we really we're really enjoying it here right yeah so it's a it's a great house and I hope to stay here for a little while longer I'm not ready to move again the packing up and move out process was just such a headache that I don't repeat for a few years but I think that's pretty much it leave me a comment down below if there's anything that you would recommend or any piece of furniture or item that you think would fit in particularly well anywhere over here and yeah thank you so much for watching catch you in the next one peace