Okay, thank you. I changed the title because I think this title goes better with the day because all my talk will be around the conjecture of Mazer and also a collaborator of mine is a recent student of Alexander Gontroff, Mr. Dimitrov. Okay. So first of all, the motivation is, let's start with Fulton's theorem. In my whole talk, I'll fix G to be an integer, at least two, and C is a smooth, projective irreducible curve defined over Q bar. The motivation is, first, Fulton's proved in 1983 that if C has genus at least two, defined over a number few k of degree B, C has finally many rational points. So this is the model conjecture. This proof also gives an ad hoc upper bound on this number. It's on G and D and the Fulton's height and about Jacobi rank and so on, but the bound is really not explicit and ad hoc one. Around 90s, Voita gave a new proof to Fulton's theorem using our club geometry and more in the sense of Dufontine method which was simplified by Fulton's and further simplified by Bombieri purely in terms of Dufontine geometry. It's a new proof of the same result, but also at the same time gives a rather good bound on it. So the bound is in terms of the genus, the degree of the number field, the Fulton's height of the Jacobian of this curve and the model weight rank. And the model weight rank. All these numbers are explicit in this bound and later on, the bound was improved by David Philippon and Heimann and so on but it remains in this term. So this is in theory, this kind of a computable expression. Yes. It doesn't involve the other and the height of the points is not known to be. No, no, no, no, that is the effective version but no. Mayser asked the following question. First of all, in 96, this is a conjecture. Later on, it was simplified. I will state it in this most general form, maybe here because this is the conjecture I'm going to study. Let's take any algebraic point P0 in C and let's consider the Abel Jacobi embedding via this point. We take any finite rank subgroup of JQR say of rank R. We hope that the cardinality of the following points, the points on C intersect this finite rank subgroup. It has a sort of not really uniform but it's not uniform only in terms of rank. So this bound does not depend on the choice of the point, first of all, and also it does not depend on the finite rank subgroup as long as the rank is fixed. This was the original conjecture made by Mayser. Afterwards, there were some simplified version. I'll talk about some. My main result today is this is a forthcoming paper with Vesselland Dimitro and Philip Habegger, is that we can prove this inequality for large curves, large in the sense that for any fixed G there exists a number depending only on G such that as long as the height of the curve is larger than delta, then this inequality holds. High of the Jacobi and all the curve. Some people also define it as the height of the curve. Some immediate application of this theorem, first of all, particular cases, if we take P0 to be a rational point and gamma to be JK, which is a finally generated subgroup, then our bound here will say that the number of rational points is uniformly bounded in terms of G, D, and the rank. Here, the dependence on D is really artificial. It is there because we have this restriction of large curves. And D is here only because if we fix D, then there are only finally many curves with small heights. So the dependence on D is only here. And this is actually another version of Mayser's conjecture. Secondly, if we take P0 to be, say, any q bar point, but gamma to be the subgroup of torsion points, right here, we only take finite rank subgroup. And this is a rank 0 subgroup. Then we have a bound that I'll just use Cq bar tall to denote it. But this really depends also on the P0 we chose at the beginning. So this is the size of the torsion packets. If you know the terminology, it depends uniformly on G and D. Again, this dependence on D is really artificial. And of course, it shouldn't even be there, especially for the second one. And currently, no results for the first one. In this direction, David and Philip Bohm proved it for the case when J is a subrarity of copies of the same elliptic curve. And they also find some examples that David Nakamaya and Philip Bohm also produced some examples of families satisfying this. And for the second one, there is a recent paper of holy Laura DeMarco. And yeah, they prove this for G equals 2 by elliptic. But their result is stronger than ours in this case, because they have no dependence on D. They really prove this. Otherwise, there are some results in these directions using the Shabotikol man method. But all of them, of course, there is some rank condition on the curve. Now let me quickly go through the method of Umberi faultings and Vojta to see some input we need. And our proof is based on the second proof of model conjecture. Recall that if we have J, the Jacobian of C, on J, we fix a symmetric ample line bundle. Then this L defines a function from the q bar points to non-negative real numbers. It's called the Nehontate height function. It satisfies two properties. First of all, this function defines a quadratic form. Simply in this term, if we take the multiplication by n of the height, it is n square of the height of the original point. And secondly, it vanishes precisely on the torsion points. Now, if we want to fix our finite rank subgroup gamma in Jq bar, we can always do a map from gamma to gamma tensor r. And here in this step, this is almost injective except that we lose all the information for the torsion. But here, we also lose all the information for the torsion. So actually, in terms of the height function and this mapping to gamma tensor r, actually we are losing the same kind of information. So we can use it to study this. And it is known that, first of all, this is just rr. And hl hat square root defines a norm on it just by the linear extension of this function. All right, now we divide gamma into two sets. For this first simplicity, I'll just present a case where gamma is finally generated. For example, the rational point case, just for simplicity, gamma finally generated. Then in this case, when we embed gamma into gamma tensor r, we obtain a Euclidean space here. And gamma will become a lattice in this Euclidean space, normally Euclidean space. And the division and Bombardier-Foltings' Vojta and their method, they divided the points in gamma into two sets, which we call the set of small points and large points in the following way. They found a c0, I'll explain later this number, and divide it into small points, which are the points with height smaller or equal to c0. And the set of large points, so you may guess, is the points with height larger than this number. OK, now we have, say, this is the origin. Say, this is the origin. And c0 is here, the radius. Now, the set of small points is here. The set of large points is outside this ball. Well, for modell, it is immediate that the set of small points is a finite set, if I restrict to finite generated group, just because it's a lattice and contained in a closed ball. So it's automatically finite. So for the sake of modell conjecture, we don't need to do anything for these small points. But then, bounding the number of large points is what we need. And this is what is done in the method. So theorem, this is Vojta faultings from Biele. And then, it was later on strengthened by Davis-Fiery-Boum and Heimann that the number of large points is uniformly bounded in terms of, I think it's 7, I'll just write it as c prime g1 plus r, c prime g r, sorry. The number of large points is already uniformly bounded. Actually, this number is g times 7 to the power of r. But yeah, let me state it in this way, every term is. And Davis-Fiery-Boum and Heimann they also proved this result for finite rank subgroup so that this part actually works also for finite rank subgroup. And they prove it using the Manthold gap principle plus the so-called Vojta inequality. So this is a review of the Bonviere faultings Vojta method. It tells us that now, in order to study measures conjecture, we only need to study the set of small points, which for modell is the automatic thing, but now it's the hard thing. And they found this c-node. So fact, the property of c-node is that it is linearly in terms of the faultings height of the Jacobian. This is how they constructed the c-node. The whole point is to find a good c-node such that this Vojta inequality holds. All right. OK, now we see that we want to study this set of small points with height bounded linearly in terms of the Hfj. And we want to bound the number of these points like independent of this Hfj. So what we do is, so it suffices to prove the following thing. Small points are far from each other. Or in mass terms, if we have any two small points in gamma, actually, we will prove it for any points in j cubed r. Then, their distance defined by the Nehontet height called Nehontet distance is uniformly bounded in terms of g and Hfj. And we cannot prove it for the moment, but we can prove it plus some number. Let's say devil not g. And this number is why we are restricted to the large curves. I'm sorry, but is it a male number? It's just a number, depending only on g. This is what we can prove. But actually, if we can prove this term, then we prove the four measures can get her in it. You were right to write it clearly. What we can do is this. Everything is positive here, but not negative. Everything is positive. But yeah, everything is positive. This may be negative. This is the problem. This is the reason why you are limited. Otherwise, you will have a bound. Yes. OK, so. Just height can be zero points. It can be zero. So that's why this bound. You mean different small points? Difference of small points. He is different from you, I suppose. It could be the same. If it's the same, then this is zero here. And it just says that the height of the curve is small. Because this is a negative number, I'm sorry. If this is zero here, then we will have zero greater or equal to a positive number times hfj plus a negative number. Yes. So that means the height hfj here is small. I think you really do want to take different points. Otherwise, you'll just conclude by taking p equals q that's behind us. Oh, OK. I see your point. Yeah, sorry. But what I want to say probably is that p minus q could be torsion. p minus q could be torsion. Yeah, thank you. Yeah. Yeah. But actually, what we proved is not just for gamma, but really for Jq bar with this term, so that we're done. Let me write down exactly what we do. We proved the falling height inequality for one parameter families. And this is a theorem. Yeah, because it's a project, it concerns several papers. So I will make it precise. Which papers are published? Which are preprints and which one is forthcoming? This one is a published one this year. It basically shows that this is true for one parameter families. So let S be a curve against smooth but not projective curve defined over q bar. Let A to S be an abelian scheme of relative dimension G, and L be a symmetric, relatively ampulline bundle on it, so that now we have a fiber-wise defined mayhotate height. Let X be a irreducible sub-variety, dominant or not. It doesn't matter. Close irreducible sub-variety. Then, OK, this is the setting I may want to use. You said you expect this to hold without the extra term delta not G? Yeah, but under certain conditions. But then it means that there is no difference which is a non-zero torsion point, or it seems too strong. Yeah, maybe I shouldn't say that, because when we put everything in a family, we see exactly what we expect from us. When we put everything in a family, we see more clearly what is expected. But here, I want to explain the idea of what should be proved to get this result. So you don't claim that you think that that's true? No, no, no. I don't claim that. I don't claim that. I just say that if that could be proved, then the conjecture is solved. All right. Now, we assume that this X satisfies some condition which I will call non-degenerate. I'll explain what it means later in my talk, non-degenerate. Then we have two conclusions. First of all is if X is non-degenerate, then we sort of can compare two height functions. The first height function is fiber-wise defined. The second height function is completely on the base. Write it. There exists a number depending only on X, and it's RST open non-empty subset of X such that the following inequality holds if we consider the two height functions. First of all, let me first of all finish this and then explain what I mean. First of all, my small s is just a point on the base curve. So this function, it actually means hf as, but I write it in this way to emphasize the fact that this function is completely on the base. It does not really depend on the fiber. And this point P lies in this RST open dense subset, but it's also on the fiber over s. So this height function is really the fiber-wise defined Nehontate height function I talked about, which is above. So the left-hand side is something depending purely on the fiber. The right-hand side is something depending purely on the base. After all, there's no relation between them. We can't expect to have a bound. But here, what we proved is that if we assume, if we restrict to a sub-right, he's satisfying some property, then these two heights can be bounded. And of course, in order to make this useful, we need to say that what is non-degenerate? There is an original definition, but it's ad hoc. So here, we also prove that we also give a criterion of this non-degeneracy. X is degenerate if and only if it's very rigid from geometry. It's simply the translate of an abelian sub-scheme translated by a torsion section done by, I would say, something constant. This means that the abelian scheme may have a constant part, it's a trivial part, and then the other part of X is just a trivial product up to a finite covering. So in practice, we prove that in practice, we can often check this condition to determine whether X is degenerate or not. And then if it's not, then we can apply this. I'll give you an example. X is a dominant over S. Yes, it's better to assume it, but otherwise these whole things work because this is just one number. Let me assume this. But it is not necessarily absolutely reducible, only irreducible? Absolutely irreducible. Everything is defined over Q bar, so it's absolutely irreducible. No, in the total space, in general fiber it is... No, it's not as well. So when you add torsion... This also may not be irreducible. I say abelian sub-scheme, but of course I mean up to a finite covering with base change, then it's abelian sub-scheme. I mean the finite cover of S. So up to a finite cover of S is... Yes, yes, yes. Okay, up to a finite covering of S. So what happens if A to S is just a product and it would seem that the two sides are independent? Then if X is not a constant, then they are not independent. Sorry? If X is not constant, then they are not independent. Then there's height in what it holds, if and only if this X is not a constant thing. So this result, I priori, also applies to constant abelian scheme. Yes, it does. The example is this is a preprint with three authors. We proved this conjecture that the theorem for one parameter families. Let's have a look. Now suppose we have a one parameter family of curves of genius at least two. First simplicity, I'll assume G is at least three, just for some reasons we will see. Then we consider its relative Jacobian. Well, there's no canonical way to embed C into J. But because actually here, we don't care about the actual heights of those points, but the distance of those points. So the height of the difference of two points. So we only need to care about this subrarity, C minus C in J, which is well-defined because it can be defined as the image of this fiber product to J, where this is fiberwise defined by sending PQ to P minus Q fiberwise. So we don't need to even fix a section for the base of the abel Jacobian belly. Okay, let's call this one our subrarity X. And of course, this will be the A over S. Then the assumption of G at least three, this will imply that X is also a non-associative family. Then this assumption will say that X is non-degenerate, just by computation by hand. X will be non-degenerate by this geometric criterion. Now we can apply this height inequality. Then if we take two points on the same fiber of C over S, it's equal to this C depending only on X, so S, so C, and so on, times the faulting height of the base point. Of course, for PQ satisfying those conditions. But this is an open condition for the rest of points. For the rest of points, we use those packaging arguments. But at least now we see that because this number, it depends only on the family, not each individual fiber. So we can see that now we really have this result. So the height of the distance of any two points, algebraic points, are uniformly bounded below linearly in terms of the faulting height. And this is what we want to prove this. Yes, P is different from Q again, thank you. So this tells you that if the difference is torsion, then you'll get still some consequence. Yes, if it's torsion, then the height of the curve itself is bounded above. And the P and Q are small, no, they are small. Then the curve is small, say. But you don't say that if there is a small point, the curve is, you should only say there is a torsion point. The torsion difference of small point, the curve is small. Yeah, well, if here P minus Q is a torsion, then the curve is small. Yeah, so that it can be negative. But I mean, we can take different normalizations to make it positive. There are different normalizations. Some convenience of normalizing so that this is positive. Actually, yeah, you might talk about things how it will always be positive. But the thing is, actually, we prove this purely in terms of height machine. So it's OK to add a number to this, or 1 to this height function. And this basically proves the theorem for one parameter families up to some details for the packaging argument. OK, now, I hope I convinced you that if we can generalize this height inequality for arbitrary families of a billion varieties, then we can prove the theorem. And then next comes the generalization. And to do that, I will need to explain what is this net degenerate. You will see that it comes from geometry. To do this, because we want to solve this conjecture, we can just work with the universal curve. And this is the modular space. This is the universal curve. It does not have a section, but still, it's a universal curve. Here, I'm taking some level structure. And again, I have the universal Jacobian. And I can take this map to construct a CG minus CG inside J, which is well-defined, which is well-defined. And furthermore, I would like to embed everything in the universal a billion varieties, using the Torelli embedding to embed it into the modular space of a billion varieties, principally polarized modular space of a billion varieties. And here, there's one to curly AG, which is the universal family. So in my talk, I will always work with this system. I have the universal Jacobian, and this is the universal a billion varieties. And this is the map pi. So you fix a level structure? Yes, yes. Because actually, in my talk, everything is for height machine, height inequality, and so on. So taking a finite covering, it's always allowed. It doesn't do it. Yes, yes, yes. So this is the object we will work on. And then, whatever sub-variety we take will be a sub-variety of the universal a billion varieties, but it might not be dominant. Most of the time, it's not dominant to the base. Reducible sub-variety. Again, geometrically reducible sub-variety, defined over a Q bar. And furthermore, we actually have another object in this picture that is the totological line bundle. It exists just because the modular space parameter tries not only a billion varieties, but each a billion variety is equipped with a polarization, a principle polarization. So this is the totological line bundle on the universal a billion variety. We can take the first Shen class of this line bundle. This non-degenerate condition, well, this is not the definition we take, but it's really the geometric meaning behind the definition we will take in a few minutes, is non-degenerate if x is called non-degenerate. If we take this first Shen class, we restrict it to x, and we take its correct wedge power. It is not always zero. It is not always zero. Well, one thing about this form is that this first Shen class is always non-negative. This can be computed in explicit or coordinates, and this is done by, I would say, Mach, 9 in Mach. So this is always non-negative. So saying that this is not always zero means that it's positive somewhere. At one point, that's enough. Yeah, because it's a one-one form. We have not come to the boundary, so it's really a form. Of course, it's hard. The geometric meaning of this will be then, well, if x is non-degenerate, then sort of if we want to integrate this thing over x, we will get a positive number. So that is almost equivalent to say that this lg restricted to x is a big line bundle, except that now nothing is projective. So it's really the geometric meaning, the geometric background behind. But of course, in order to get something, we need to be more precise. But this is our idea. Our description is as follows. We use the so-called Betty map, introduced by Daniel Bertrand, when they studied other kind of different time problems and like the intersection problems. The Betty map is defined as follows. Here we take the universal identity. I want to take its universal covering. Well, we know that we take the Ziegler path space. It uniformizes in the complex category, the modular space. And here on this AG, we want a similar thing. We want a similar thing. And it can be defined as follows, universal covering AG. I'll call this one, I was in very bad notation. I'll temporarily call it yg, y2g. First of all, it's defined as follows. First, as set, or as real algebraic object, it is r to the 2g times the Ziegler path space. And then we want to give it a complex structure. The complex structure encodes the complex structure of every abelian variety over this modular space. So it is actually given in the following way. Here we take any vector, AB, both in RG and a matrix, G by G matrix. We send it to A plus CBZ. And this is, of course, a bijection. And it's also automorphism as real varieties, real algebraic varieties. But the complex structure should be this one. Then this is the universal covering. I'll use this one. Now we have y2g covers AG, which maps to HG plus. This is just a trivial product with some twisted complex structure. All right. Now X is in AG. I'll take an irreducible component, irreducible in the sense of analytic geometry, component of u inverse X. Then this is going to be in this uniformizing space. This is going to be in this uniformizing space. And the Betty map is something very, very simple. It's just y2g when we write it in this way. Actually, it projects the first factor and the Betty map, essentially, is the composition. This is the Betty map. Well, this is something very, very simple, but in close, for example, we can really read off this condition completely using this very simple map. So I'll give the actual definition which we have. X is non-degenerate if and only if, or if and only if. This is definition lemma, maybe. B, restricted to X2, has full rank. The generic rank of this map is twice the dimension of X. So this is because this 1, 1 form, it defines a point-wise, a symplectic pairing. And the kernel of the symplectic pairing are precisely the fibers of this map. This can be computed explicitly. And then saying that this is non-zero precisely means that there should be no direction along X not vanishing along some positive dimension of the fiber. So this is precisely the definition. And also from this new definition, we do know what kind of sub-right is degenerate. So from this new definition, we have that X is degenerate if and only if for any X2 in this X, the big X2 component, it lies in a fiber. So the dimension of B inverse BX intersect X2 is positive. The dimension of that is positive. By the way, the fiber, the map is just real analytic. It's just real analytic. It's not complex. The fibers are complex analytic. And this is used in the verification. Yes, because it's pure. It varies homomorphically. Thank you. And then this says that now we have actually a curve in HG plus, a complex analytic curve such that if we take BX times this curve, this is inside R2G times HG plus, it is contained in X2. It's containing that for each X2. And then it simply means that X2 is covered by such things. So sitting there just a local curve is a... Yeah, but it's complex analytic. You can use analytic extension somehow. Yeah, but it just defines a small neighborhood. Yes, yes, yes. Our priority is only a small neighborhood. Yes, yes. And here, I want to say that now then this hopefully implies that if X is degenerate, then it's the union of such things, but I can also take the diracic... Oh, sorry. So X, now when we pass to the universal abelian variety, because X is algebraic, so I can take his diracic closure. So that now X, if X is degenerate, then it is this ad-hoc union. It is this ad-hoc union. Actually, the other direction also holds by some... a lot simpler reasons, but maybe I... If I have time, I'll explain it, but the other direction is the simple... it also holds. All right. Now we want to study these things now. Okay, now the question is to study these diracic closures. What properties should they have? And this is the next part of my talk, maybe number three, functional transients. I'll start with the weak X-chanu statement. This is actually a sort of statement. It says the following thing. If we have two varieties, two algebraic varieties, and this is more or less... this is subjection. This is usually the... the usualizing map linked by a trans... an analytic map. So both omega and s are supposed to have some algebraic structure, but Q itself is very far from being algebraic. It's highly transcendental, but it's only analytic. Then weak X-chanu says that things here and things here, they should intersect transversally, except if they... except if they actually intersect in a smaller base, meaning that if we have z inside omega irreducible, complex analytic, again irreducible in the sense of complex geometry, then the dimension of z bar inside omega plus the dimension of Qz bar inside s is greater or equal to the dimension of z plus the dimension of this Qz Bzar. Let me explain this term. So we have a bioalgebraic system here, and we say something is bioalgebraic if it is algebraic here and here. That is the Bzar thing. So this is sort of the bioalgebraic closure. weak X-chanu says that this thing, it is in omega, an algebraic thing in omega, and this Qz-zar is an algebraic thing in s, and the dimension of their sum is at least the dimension of the z itself plus the Qz-Bzar thing. And it means that these two things, if there is a proper bioalgebraic thing containing both, then they don't intersect transversally. Otherwise, they always intersect transversally. So this says that if we have some sort of unlike intersection, then there must be a reason. And this reason, we can write it down. This is the result we're going to use. This kind of result was first approved by ARCS, of course, to study the shadow. It can be a functional analog of x-chanu's conjecture, but it got interest again by the study of the Andriot conjecture. And it's another, it's a weaker form of a weaker form, Ax-Linderman, was proven by Klinger-Rilmore and the FIF. This is for Shimura variety. This is for Andriot. But this general form recently is proven by Mark-Pilan Zimmermann for the modular space. And this paper is published. It has two extensions, two directions of the extensions. First direction is by Baker and Zimmermann for variation of pure hard structures. This is also published, and this is used in the recent proof of Lawrence-Tanban-Kentash proof of model conjecture. Actually, it's used for the high-dimensional, high-dimensional case. Another extension is done by me for this universal Abelian variety. It's a mixed Shimura variety. It's not pure anymore, but this is a preprint. And in studying this question, studying this y, it is this form of use. Now to study that y over there, we see what condition it satisfies. We apply the weak Ax-chanu to... Here, z will be r times C-tutor. It is also complex analytic, as we said here. And of course, to this system, to the system y2g2AgU. Well, then first of all, the first term is dimension z-bar, which is the dimension of r times C-tutor zar. This is the first term. The second term is precisely y. And the third term is 1. Well, because it's a curve. The third term is 1. And the last term is y-bizarre. Now, in this inequality, actually, we know many things. This is what we want to know, so we don't need to do any operation on it. This is a number. And this is a biogereg thing. Well, in biogereg geometry, usually these kind of things have very good geometric interpretation, so we don't need to do much about it. So the only thing we need to study is the first term. Well, a first step to do is that this is our C-closure. Although this is not trivial, but it's not very hard, actually certain things commute. We can first of all do the C-closure of C-tutor completely in Hg, and then do this product. They are actually the same. We can show that. Then the equality becomes something purely in the base plus dimension of y is greater or equal to 1 plus dimension y C-bar. Okay, now let's study this. Well, by the choice of C, we know that y is r times C-tutor and take the zar C-closure. So of course, this thing is smaller or equal to dimension of pi y bizarre. This is smaller or equal to pi y bizarre. So now we have the equality being dimension of y is greater than, I removed this one, but then replace this by strictly larger than dimension y bizarre minus dimension pi y bizarre. Okay, this is actually the eventual form we want because both things have very good geometric interpretation. I can also tell you what is this difference. This difference has an even better geometric interpretation. Say here we have our universal Abellion variety over the modular space and this is our pi y bizarre. The y bizarre is something over that and it is part of it. This part, it is precisely up to a finite coloring. An Abellion sub-scheme translated by a torsion section then by a constant section or isoconstant section. So in some sense it is the smallest Abellion sub-scheme containing this y. It is something like that. So now y is something here. Say this is y and if we restrict the orange part to the yellow part then y will be contained in no smaller Abellion sub-scheme in this whole thing. Then dimension y being larger than this difference means that the dimension of y is larger than the relative dimension of this Abellion scheme and if we look at the definition of the Betty map, of course the rank of the Betty map cannot exceed twice the dimension of the original variety. So that's why as long as y satisfies this then of course y is degenerate and if x is the union of such things then x must be degenerate. This is why I said that the other direction is rather easy to prove. You say it's in a billion sub-scheme and sub-scheme over what? Over a sub-variety of A.G.? Yes, over its base. And this glass section? The torsion section over its base and the constant section over its base. Now I'm restricted everything in the base. The base is fixed. It's translated by torsion section of this restriction and then by constant section of course when I talk about this I always mean up to a finite covering. So the torsion section is actually constant after the finite cover? Yes, it is. I just want to separate torsion and constant but otherwise I can say just constant. It's fine, it's fine. But because in studying this mixture or whatever this stuff, torsion and constant, they are really very different things so I like to separate them. One of them is special, the other one is wicked special and so on. So, okay. Now using the mixed-axe channel we can show that X is really the union of something and still an ad hoc union but each member of this union have a better interpretation. It satisfies some condition which is more or less standard in electrical intersection theory. And then next step is really do electrical intersection theory. So next step do electrical intersection theory. Again, this is basically says that the maximum Y is satisfying this condition they come from finally many of families. This actually goes back to even Bogomolov for a billion varieties and then Emmanuel generalizes to Shimura varieties for weekly special sub varieties and on the other hand when studying those electrical intersection Zuberpin conjecture Habegger and Pila introduced this weekly optimal thing and Dao and Ren proved it for Shimura varieties I proved it for mixed Shimura varieties so we can show that those maximum Ys come from finally many families. So eventually this ad hoc union becomes a finite union so that for any X this union will be ZRC closed and also we can prove the criteria in one this ZRC close-up variety is the whole thing doing this electrical intersection stuff I don't think I have time for it but just back to our counting problem back to Meisler's question of course as I said we want CG minus CG to be non-degenerate unfortunately it's not true for stupid reasons because the dimension of this is too large compared to G so this is not the correct X we take the correct X we take there are two ways I'll take the easy way it's just a fiber product with piece I think here I can write 3 but I really should write 3G minus 2 okay for some reasons for some induction reasons and of course here assuming that G is at least 3 if we take this power to be 3G minus 2 then X is non-degenerate so that we can prove this height inequality it says that okay we cannot prove that each pairs of different algebraic points are far from each other but we can show that around each algebraic point are at most 3G minus 2 points which are not far from it and for the purpose of counting that suffices so for each algebraic point at most 3G minus 2 points close to it close means that the distance is smaller than the linear function on the Fulton's height then for the purpose of counting we are done just multiply everything by this number which still is explicit on G I think I will stop here thank you well thank you are there any questions could you repeat why you have to rule out the case G equals 2 oh actually it's just because when G equals 2 this will be the whole thing whatever power you take it will still be degenerate the way we yeah yeah yeah the way we solve this problem actually this is not the actual thing we take we take I know this the map we take this is K so this is K we take is P1 minus P0 the map we take is actually this one for this as long as K is at least 2 or 3 then G equals 2 will also be solved at the same time for one parameter families it's almost equivalent to take this one and this one doesn't matter but actually when we work on this even for G at least 3 it's simpler to take this one just for the proof with Joe Aris and Luceca Forazzo he has been even more optimistic about rational point optimistic I don't know for that thing I really don't know because it's a rank stuff I can I can do nothing with the ranking but I have a question about it so you do it for finite rank subgroups of J, Q bar and of course also J torsion what about a sum of finite rank and J torsion I think it's a finite rank right what it's a finite finite rank Q rank oh no no I don't mean that I mean the visibility of if I allow myself to divide finite rank subgroup by integers and I get a oh that's actually what I mean by finite rank it's Q yeah yeah yeah it's Q rank yeah yeah yeah finally generated I mean that it's a Z module but finite rank is really over Q yeah yeah yeah because in the proof actually for the large points they already solved this problem and for the heightened quality we don't see we actually don't see this gamma thank you


And on to other issues, the governor of Oyo State, Shai Makinde, has condemned the recent violence along Iwar Road in Ibadon. The incident led to the deaths of two persons. The governor toured the affected area and said the government will bring the perpetrators of the act to book. Makinde described the occurrence as unfortunate and also ordered investigations into the violence. Hello, hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


I'm doing a project with 1800 tequila where I'm repurposing raw pieces and making a unique one-of-a-kind Outfit for an athlete for the red carpet. It's gonna be a pretty big deal. I Only have two weeks now to put this together. So it's gonna be very tight So pretty much what I want you to be is a sexy one. I mean, I think that comes natural for you, right? Oh It's great. That's the one I think Los Angeles is having its moment. There's so many creative things here and so many creative people You can find inspiration anywhere, but for me It's it's the inspiration. I Haven't found the fabric that I'm gonna use for the shirt The blackout curtain that has, you know, just an amazing drape the color contrast. Yeah I See the world from an artist's point of view I'm always looking for the beauty in something and what's beautiful about the cupra is that it feels like a silky material Because the way it's processed it comes from the outer lining of the seed of cotton Gives it a very fine hand and in giving it a fine hand. It has a very fantastic drape Thank you so much. Yeah, man. I really really appreciate it initially with the shirt It was going to be long sleeve, but long sleeve really don't make sense because it'll be between 95 to 105 degrees on the red carpet So it went from a long sleeve to a short sleeve and ultimately rested on no sleeves I now hand this off to my team. We cut it and construct it Human beings were all perfectly imperfect and it's no different when you're making garments by hand The raw elements of the fabrics give the finished pieces character We all have two hands and ten fingers, but we all have different fingerprints or to crystallize that this is specifically made for you I want the client to feel like somebody receiving a billion dollar check That's empowered Thing that makes my job edifying is just what you can find in terms of fabrication You have something that drapes great and then you have these amazing Camouflage pants that have been repurposed Two pieces down one piece to go You


senior MOO3 and senior NBO3 are 2 4D correlated metals which have been used as transparent conductive oxides, TCOs, due to their high optical transparency and low electrical resistivity. In this study, these two materials were grown on CA2 and B3O10 nano sheets, allowing them to be grown on technologically relevant substrates such as fused silica. This allowed for the production of highly conducting and transparent 4D correlated metals, making them suitable for use as TCOs in the ultraviolet visible spectral range. This article was authored by Foo Tran Phong Lu, Xu An-I, Pierre-Alexis Repicott, and others.


Hey guys, welcome back to my YouTube channel. This is Daniel Rossell here in today's video I want to talk about a few resources that are out there if you're relatively new to the world of Linux and you are Delving into the sometimes fraught subject of hardware support What hardware will work with your Linux computer whether you're looking at buying a webcam or a scanner or Any other kind of hardware for that matter? You may be interested in knowing the important question of well, is it going to work one thing? I will say I've been using Ubuntu Linux since 2008 and hardware support has gotten Considerably better over that time period It's now there's now a much much higher chance in in my experience that devices are plug-and-play Nevertheless, it always pays if you're thinking about buying something whether it's a you know Blu-ray burner or as I said a scanner or especially a printer Pays to do a little bit of research just double checking if something's going to work with Ubuntu So I want to show you guys a few resources and make a few comments First comment really is that you're going to have a higher chance of support if you're using a more common formal Linux So there's many many Linux distros of course So if you're using something relatively common like Ubuntu The chances of you either being able to ask someone or find a resource for whether it'll work. They're going to be higher There's a few different URLs. I wanted to pop open to show you guys So Ubuntu has made this attempt to Have Ubuntu certified hardware Ensuring that Ubuntu runs well out of the box So this is a nice program and they're trying to sell this to manufacturers in terms of This the certification program just to make life easier for us Ubuntu users, but you can see in the desktop space They have 277 Desktops a lot of them from Dell that have been certified to work with Ubuntu if I filter on HP We can see there's a few more and they even certify them by release. So this is definitely a step forward You can see this HP micro power TC is a desktop system certified with Ubuntu 18 04 so this is a Certification program being rolled out by Ubuntu the categories up on the website currently are Ubuntu certified desktops Laptops, which is super useful. I've never looked for an Ubuntu certified laptop I've also never had problems with Ubuntu, but I have had problems with Debbie and in any event You can I get desktops and laptops pre-installed with Ubuntu? I think this is definitely a really good idea and you can see for the latest LTS 20 04 There's 284 Ubuntu certified laptops at the time. I'm recording this video, which is 6 of June 2022 Ubuntu certified servers Ubuntu certified IOT devices Ubuntu certified systems on a chip SOC's and They're trying to sell this to vendors as well So that's a nice program and this is the curve. This is another link here about their Ubuntu's Ubuntu's programs you can see the current programs called Ubuntu certified They did have Ubuntu friendly, which was kind of a previous attempt at this. You can see it was wrapped up in 2013 They have a wiki as well, and it's a bit more extensive than Than the previous page. They're not certified But for example, if we pop into the webcams category here We can find let's say for instance, we want to filter on logitech webcams and there is a wiki. It's not beautifully Organized I'm not sure it's even 100% complete. It's very out of date You can see we're going to Ubuntu 10 here Ubuntu 6 So it doesn't look like this resource is very well maintained unfortunately last edited 2012 So that wiki isn't super useful Moving a little bit outside of the official Ubuntu ecosystem here There's a website called linuxhardware.org I'll put a link in the description and I also will just do my usual Technique of flashing it up here on the screen. So that's the URL linux-hardware.org This is a pretty useful. This is obviously not run by Ubuntu Or anyone like that, but it's actually pretty seems to do a better job at Listing so if we go into you can look for parts here vendor find a computer and They even list webcams and sometimes you'll be searching for a computer and it'll show up Next resource for you guys is to show that I wanted to highlight is the linux hardware subreddit So it's our linux hardware and there is 66,000 people on the sub so it's not a tiny subreddit Sometimes you just can't beat Human intelligence so asking people. Hey, does this webcam work or does this motherboard work etc? So you can ask I'm not sure there's a flare for saying does it work? So I try to periodically post your saying this works etc But this is a good place if you have questions you can pop over to linux hardware and ask We've already looked at that the next way I'm just going to take an example of the Logitech C930E Camera right so then you've got googling which You know the good old Google method type it into Google and for instance when I typed in Logitech C930E We saw that it had an entry on hardware for linux, which I showed previously So it's listed there and it'll show you which kernel drivers you need and the the status that on Ubuntu 20.04 it was detected Three different probes and you can even go into the individual details so that to me that looks like it works I own the C920, but I wouldn't doubt that usually Usually well-known popular webcams if you're using a pretty standard type of linux like you know Ubuntu You can see all the distros here fedora majoro arch usually mint nowadays. It's getting good. We're not not really Running into problems so frequently, but there is these databases the second thing Let's go back to this Logitech C930E webcam, right? So I buy a lot of stuff from Amazon or I use Amazon as a Lookup tool to do my product research So what you can do on Amazon is scroll down now You'll see for instance this webcam doesn't mention Ubuntu supports It just mentions Windows and Mac. You'll see this very common. The vendors are generally a bit reluctant because there's so many Linux distros. I think they're reluctant to say this works on Linux because it might work on Common distros, but then there'll be one distro won't work on and then you've got a bunch of angry customers Who are like it says it works on Linux? So my experience is vendors are reluctant to put that Linux stamp on nevertheless what you can do is Type into the review section Ubuntu and you'll find people commenting just a purchaser So you can see here Is this plug-and-play with Ubuntu and this guy says I'm using it with Ubuntu every day I wouldn't expect this not to work. You can also try the Linux keyword for other distros This guy's using it with Linux Mint or you can type in for example arch. I mean you might have more More trouble there as you can see so that's kind of the main ways in my experience I've been using the internet to research whether hardware works works with Ubuntu Sometimes drivers are a bit buggy, but for the most part these days I'm finding increasingly stuff is plug-and-play and works nicely out of the box Hope this video was useful and thank you guys for watching


The web of things, internet of things, how they can provide services to the users on the web is very interesting. Let's look at a scenario, particularly our smart home, which is going to have certain services, which can be tailored, which can be revoked, provisioned, built, and then we are going to identify interesting requirements that we'll discuss in due course of time. So we call it the smart home scenario. We look at how a smart home offers services through the WOT service model. First of all, as in WOT, all the interfaces are based on HTTP, understood. But the possibility of having the proprietary protocols and the broker based protocols makes the WOT service model more powerful. As a consequence, the cost is these possibilities allow or trigger certain degrees of service qualities and traffic patterns. It means that we are going to expect non uniform and heterogeneous or traffic behaviors and the service qualities which are coming for different applications on WOT from our different physical devices. So what are the WOT devices as in smart home starting off from the very basic things such as electric heater or gas based heating system with access to the room heating or the water heating like geysers. Then we've got HVAC, the humidity ventilation air conditioning. We have the entertainment system like LED TV. We have the entire household lighting system, including the lights of every room, even down to the refrigerator and microwave oven lights. Then we have cooking system, including the oven, the cooking range. Then we've got surveillance cameras deployed across different ends of the household. Then we've got the management of the main gate, the front door and the wall mountings, including paintings and the pictures of family on the wall. Then the vacuum cleaning system. In this case, we can think about a smart robot, which cleans only the relatively dirtier places over the last 24 hours. Then we've got the temperature control system of the entire house depending upon what is the temperature outside. We always had such systems like we had the Honeywell system or we have the Nest system provided by Google. So if you look at these two protocols or these two technologies which are offered there in the market, you'll see that one of them is heavily proprietary while the other is based on the internet infrastructure. So the WOD services are respectively created for all these things. For example, we need to have entertainment system, entertainment service, the heating service, heating, cooling service. Then we've got the lighting control system. We've got the visitor presence service on the main gate and the front door that works in collaboration with the cameras. So we can think about the smart home as a very interesting, intertwined set of services. So at the device layer, we've got the devices themselves. Then we have the broker. The broker comprises the adaptation layer and the service layer. So you can see that we've got the surveillance service and refrigeration service. Now these services are accessed, edited, manipulated through an app. This could be an Android or iOS app. So having such a complicated scenario means that we are not talking to the house alone. We are talking to certain power generation company, certain gas provisioning company, some security system or the security infrastructure. We are talking to the weather forecasting website. So it means that the smart home application is a very good starting point to look at the requirements. For instance, who owns what and who gets to have access to what data type and what kind of traffic pattern would be generated. Then how are you going to manage the pricing of it? How are you going to monetize it? How are you going to deliver it? And how would you see that certain quality service parameters are met? And are you going to charge it in a certain pre-paid, post-paid model? Are you going to offer certain subsidized bundles? And what is going to be your billing mechanism? How are you going to actually collect money from the service consumers? So now the smart home scenario sets the right basis for further discussion and follow up on the charging, rating, the business aspects for the IoT environment.


Welcome to Geobiology. Geobiology is the study of how earth and life interact and change each other. So earth provides the boundary conditions for life, it provides the resources for life to exist. And life performs reactions, organic chemistry reactions that use those resources and change earth. The surface of the earth is really shaped by the life that's on it. And as life has evolved through time, it has changed the surface of earth through time. And that again feeds back into the life that can exist and where it can exist. So geobiology is really the study of the planet earth, the life on it and how the two interact through time. So in the course of these lectures, we're going to cover a lot of different concepts. We're going to start with the idea of feedbacks between earth and life and then talk a lot about the different types of interactions different species have with each other that helps shape ecosystems and really structures that interaction between planet earth and life. From there, we'll go into more detail about what life is, what it needs and the life that's specific to what we know on earth. And in that particular case, we'll be talking about genetics and enzymes and how life evolves through time by the selection of the function of those enzymes and other components of life that's determined by the genetics. Then we'll get back to those interactions between different species and life in talking about ecology. And then we will come back and really talk about some of the details of how life and earth have interacted through time and continue to do so today. So I'm really looking forward to sharing this information with you. And it's one of my favorite topics that's something I've devoted my career to. And it's always wonderful to share what I've learned. And I hope you enjoy it. Thanks for watching.


The making of The Never-Ending Story was a real rollercoaster ride, a wild rollercoaster ride. This movie was something that everybody could relate to. One of the greatest films that has ever been made in my opinion. Business to place where The Never-Ending Story were shot. The story itself is really, really profound and deep. It gives me goosebumps even to just talk about it now. Never-Ending Story. He's got that X factor. Noah, he's had a lot of trauma from the business. I had to learn how to walk again. Anybody who's been famous since they were a childhood adult, I can assure you that they're not in reality. Yeah, cash literally saved my life. I just lost my mom last week. Dog was much harder. You're looking at 30 years of neglect. I know Michael Ender didn't like the film. We're paying his ego. It became even bigger. It's more simple, it's more effective. And the painting is telling me what to do. The horse was really going down there. I tell people I got a fishbowl with dollar bills, please take one because I know you went to therapy. This is the first time I'm back in Munich since I was 10 years old. So many of us as children of the 80s are now adults. We're showing it to other generations over the years and it just remains a part of our lives. People park back to that time in their life before responsibility, before, dare I say, some pain. It's perfect. But that's another story.


You know, it's never an easy task to discuss tough issues, sensitive or divisive issues. But leadership is not found in the avoidance of the difficult matters, but in confronting them head on. That is what Forbriton has pledged to do, and that is therefore what we will do. In our current era of censorship and de-platforming and shouting down, some issues have been buried, deep underground. But that has not made them go away, instead they bubble under the surface, ready to explode, an explosion that may well cause more harm than the issue itself. I'm referring to the issue of demography, or more specifically, the demographics of the United Kingdom, today and tomorrow. Why must we at Forbriton take hold of this issue now? Because it is crucial that we do not leave it in the hands of extremists, racists or neo-Nazis. We refuse to allow such people to gain ground. Britain is a civilised country, and we will stay that way. We will conduct ourselves with decency and with fairness, because to do otherwise is to betray the very values we seek to defend. Demography is not, however, a simple science. It is true to say that people have been moving around the world for centuries. We have mixed and intermingled for hundreds of years. But this speech is not a study in anthropology, but in common sense. Common sense dictates that we only live in the world as it is today. We are under no obligation to pay heed to society's centuries ago. We are only obliged to pay heed to our lives today, our time, our era. Because this is our time, this is our era. We can only live our lives today. We cannot live them yesterday. Our current demographic then is the only demographic that matters. It matters so that we may understand where we are and where we are going to. As a starting point, we must accept that change is inevitable. No matter how uncomfortable some changes may make some people, change is still inevitable and we must operate with that reality in mind. The world never stands still and nor do we desire it to. We must use reality as our starting point. So let us state some realities. In 2020, there are millions of British citizens who are not white. That's the reality. They are our friends, our neighbours, our colleagues. That is also the reality. For Britain will not turn our backs on our fellow British citizens irrespective of the colour of their skin. We are clear. All British citizens are equal. All British citizens have the same rights, the same value, the same dignity. There is no compromise on this matter. Now whilst we are committed to those values, we are equally committed to speaking to and responding to the very real concerns of the native British majority, whom I'll refer to as government documents do. White British. Often upon mention of white people, some will ask, well, what do you mean by white? Well, I mean white. I mean people whose skin is white and whose ethnicity is therefore European. Once again, this speech is not about anthropology but common sense. And while anthropology may be somewhat ambiguous about ethnicity, common sense is not. It may not have been the case that Europe was white 50,000 years ago, but we are not concerned with 50,000 years ago. We are concerned with today. And today, Europe is ethnically white. For those who argue that there is no such thing as an ethnic European, I say, OK, I accept that, but only if the same applies equally and everywhere. If there is no European, then there is no Japanese, no Nigerian, no Native American and no Aborigine. If those ethnic groups exist, then so do Europeans. But nobody would dare to tell the Japanese people that Japan does not belong to them. Nobody would dare tell them that they are not an ethnic group. So let us give white Europeans the same courtesy. We do exist. If we don't, nobody does. If we exist as a group, therefore, and we do, we demand the same recognition, protections and rights as any other ethnic group, including the right to our culture and our ethnic homelands. For if Europeans are not entitled to recognition of our ethnic homeland, then no should any other group. Just as we would not tell Japanese people that Japan does not belong to them, nor would we tell them that they have no right to remain a majority in their own land. By contrast, Europeans are told on a consistent and regular basis, both implicitly and explicitly that Europe does not belong to us. But it does. And despite the global and government-funded anti-white racial hatred that is so prevalent, that fuels violence against Europeans and has resulted in countless crimes against us, despite this, the truth remains true. Europe is the ethnic home of white people and we demand that recognition. We demand also the right to remain the majority in our ethnic home, something other groups take for granted. As part of my commitment to reality, I acknowledge, as must we all, that the clock only ticks in one direction. There is no going back. So let us look instead to the future. For Britain is clear. We defend a Britain that is majority-white and we defend its right to stay that way. This does not mean that we reject our non-white brothers and sisters. It simply means that we are reclaiming our historic identity and that is our right. For Britain is committed to maintaining a British majority in Britain. This prospect does not, and I repeat, does not oblige us to indulge in crass geneticism, nor does it oblige us to seek to remove our fellow British citizens from their own country on the grounds of their skin colour. We will never succumb to such disgusting and dangerous notions. For Britain is equally committed to maintaining the culture of the British majority. For this, we must be recognised as its indigenous people and we demand the same cultural protections as all indigenous groups. Non-British languages will not be accommodated. There will be no cultural or religious sensitivities in schools, hospitals or police stations.  They have every reason to be. On day one of a for-Britain government, immigration will stop and deportation will begin. With very few exceptions, all illegal immigrants in the United Kingdom will be returned to their country of origin. If they refuse to name their country of origin, they will be held in detention until they do. For a period of five years, there will be no permanent migration to this country, no indefinite leave to remain, no more new British citizens. We will spend that time restoring order and doing all we can to ensure that those who reside in this country do so legally and who are contributing to the wellbeing of our nation. Those who indulge in criminal practices will be removed permanently. Those who engage in atrocities like female genital mutilation, forced marriage or honour violence will be removed permanently. Those who come to Britain to preach hatred against our society or to encourage the overthrow of our democracy will be removed permanently.  Those involved in so-called grooming gangs, even those who are reasonably suspected of involvement in grooming gangs will be removed permanently. There will be no more brokers sharia councils, religious slaughter and no more will the British taxpayer fund enormous polygamous families. Polygamous families will be removed permanently. In short, for Britain is not willing to agree to the status quo. We will not take this forever. No more will we allow ourselves to be reduced to second class citizenship on our own lands and no more will we look away meekly as our laws and culture are trampled on. This is our part of the world. We are entitled to say who lives here and who does not. Some predictions state that the British will be a minority by the 2060s, but none of this is set in stone. If we take these tough actions and we do it soon, there is nothing at all inevitable about becoming a minority. What we need, however, is not Nazism or fascism or racial division and conflict. What we need is a strong British identity that brings all British citizens in the same direction, a direction of liberty, truth and democracy. That strong British identity must be lifted from the values and the sacrifices of those who went before us. We will take the best of Britain's past, the best of Britain's present and we will build a British future. Our demographic destiny is not set. Our future is not written. We are its authors. So let us start today. Start now. Let us step up and take responsibility for our nation. Let's fight for its preservation, but let us never lower ourselves to fascism or racism in order to do so. Join us and come with us on this journey. Our destination is Britain, a free civilised, just British Britain.


When Secretary of Defense of United Kingdom, it's a very good staff. Thank you very much from Ukraine to United Kingdom.


So you're teaching brand new.  I'm going to show you a couple of really simple things that are going to make a huge difference and accelerate your throws training. So check it out. Everybody, it's Eric Johnson from Airtate Throws Nation. In today's video, we're going to talk about another thing on how to hold the shot. But we're going to talk about it for super ultra beginner throwers. So today, I have my trusty helper, Ella Johnson. And what she's going to be doing is we're going to be showing her how to hold the shot properly hand. So one of the things we're going to do is we walk through and we show Ella how this works. So one of the things that we're going to look at is when we're looking at this, we're putting it up on the knuckles. And notice where the thumb is. The thumb sits here. A lot of times, young throwers put the thumb on the side of the shot. So here's what we're going to do. We're going to try to put that shot put up on your knuckles. So what we're going to do is we're going to show our young thrower how to do it. So she's going to hold the shot put. And she's going to put it. See how it sits there? Yeah, exactly. Now look what she did. She naturally created that little cavity here. She's got her thumb behind the shot and her fingers behind the shot. And she's going to be developing her hand strength so she can learn how to flick the shot off. That's going to be a really important thing that you want to learn at the early stage of a thrower. So now notice how she's got the shot. So you're doing a very good job. Now it feels hard when it's out here, right? OK, so watch. Put it right up here on your neck. Easier? Yes. So it's called the shot put because we're not throwing the shot. We don't ever want to do this. And notice again, just that little adjustment by keeping the hand in the right position, the thumb automatically fell in the spot. Notice how she's got it here. She's got a nice hand position. And we want to keep those fingers a little bit more here so that the pinky is a little bit more like that. So now it's going to push off of these fingers, those three fingers, OK? And so what she's going to do is put it in her neck. And what we're going to do is teach her how to get the elbow out. And that's going to make a huge difference. So now that she's doing this, you can see that we really quickly taught her where does that shot go? Center that shot goes on those fingers. And then that when she pushes it up in the hand, it becomes more of a natural position to want to put that thumb behind the shot. Now one of the things she's focusing on is where does the elbow go? So now I'm going to have her stand sideways. And so now you can see. So if we're looking, we need to start teaching her how to get that elbow kind of back. And this is going to be the thing. So we're going to pull her little shoulder blade back. And now when she pulls that here, whoop, OK? So you can see how it's falling because she's not used to this. So Ella, here's what you're going to do. You're going to hold that shot like this.  But look at my arm. My arm's not here. My elbow's back here because we're going to be getting our elbow behind the shot and we're going to be able to push, OK? So what you want to do, sweetheart, is you're going to do this. You're going to feel it. See how your wrist has to sit more like that? See how I kind of curve her elbow?  You've got to press it. Boom. You've got to push it in there. Now, see how she's turning her wrist? This is going to be the common thing. And this is why young throwers tend to come out with the shot and they don't know how to push. So what we're going to do is we're going to have her hold here, OK? And again, notice what I'm trying to show you. So you see where we're trying to get her elbow up and her wrist sit like this. See how you're going to have to push? So that's what you want to feel. So see where your wrist position is? Now pull the shot into your neck like that with your elbow there. But see how, as I start to pull her in, she wants to drop this elbow and turn the hand like this. The hand position in here would make her want to throw it like either basically a straight-up motion because you have to have the elbow behind the shot. So this is why we're trying to teach her to keep the, this part of your arm needs to face the ground. So you do this. There we go. So let's try that again. So you're going to pull. As you pull, we're going to turn the hand sideways. We're going to pull it in. Much better job. Now she would be here. The crease of the elbow is facing down. Now when she puts the shot in her hand, she should be able to push the shot. And what we're going to do is we're just going to push it. And we're going to push it up right here, OK? 45 degrees. We're going to just kind of get it right there, OK? So let's give it a try. And what we're going to do is come over here. Keep your abs tight, OK? And what you're going to do is put this foot forward, put this foot over here, over here. Yep. So now what we're going to do is we're going to turn your hips this way. So turn your foot, get your knee up, OK? And we're just going to try to put your arm like this. And you're going to pull and you're going to push that shot put up in the air, right? It doesn't have to go far. Just push it up in the air. Nice job.


Good morning everyone. Welcome back to another vlog. I'm currently getting ready. It is 8 a.m.  I have really been loving my weekends in Texas in quarantine. I've just been really focusing on having like actual like relaxing weekends and like spending time reading, extra time cooking, like things I really really love. I think it's been nice to not like pack my weekends with things that I need to do and want to do. It's been really nice having just like weekends at home that are a little more slow. Love that. I actually am going somewhere tomorrow which is exciting. I feel like I have to preface this in every video. Texas has lifted state and home and I believe we're at a 50% capacity now. I haven't really been going many places. I might like pick up coffee this morning or something. Okay, I'm ready. Anyways, I am gonna go somewhere tomorrow. Obviously like wearing masks and stuff. It's like totally okay. Also, the way hair products have been helping my hair so much. It's really dull. I need a haircut really bad but it has just really brought it back to life. So I've really been loving that. I also need to make a video public. So that's what we're gonna do. Okay, I'm gonna make a little espresso shot. I've never made one from home but look at this cute little espresso shot glass. It's glass. I got it on Amazon. It's not the exact same as the coffee mugs that I use every day. Those are from William Sonoma. But here we go guys. This is exciting. Okay, this is gonna be interesting for sure. Blinding in my house is wild right now. Let me see if I can fix this. I don't think I can. That's not gonna do anything. It's like literally beautiful and there's always rainbows on my table. I normally chug an entire glass of water before I drink any coffee but since this is just an espresso shot, I feel like, you know, I need to go to Costco and buy in bulk because those are so expensive. Are you ready for this? Coco, are you excited for your walk? Are you excited? Oh my gosh, you look so cute today girl. Are you excited for your walk? Guys, I found an old video of Coco with the perfect hair length. I have never seen her look cuter, ever. I'm gonna start showing this photo to all the groomers and be like, why are you not doing this? So good. Okay, ready? It's hot obviously. Maybe I'll just start taking the espresso shots in the morning. Okay, that was like my strongest pod. It's not the worst thing I've ever had but I'm also like really good at doing that stuff. So shots are like my hidden talent. Wellness, all of them, you know? Well, hopefully that helps me.  Guys, I forgot to show you guys what I'm wearing. So these are my favorite shorts from Outdoor Voices. I'm sorry about the dishwasher. I tried to link them in the video I just posted today and they're not on the side but I'll keep looking and figure it out for you guys because I'm obsessed with them. I want like 15 of them. I'm gonna live in them all summer. Black tank top from Miss Guided. This exact one, which I'll link to this exact one, is great quality. It's super inexpensive. I bought another one from them and they're like two cropped and stuff so I'll only like wear it around the house. And even then sometimes I'm like wearing it in vlogs and so like I pull my pants up all the way because it's just like really cropped. So anyways, I really like this specific one. I'll have it linked down below but it's really great. I have a white one as well. And then this I'm just gonna throw over my shoulders. Very country club mom of me guys. I almost wore a visor but I didn't. I don't have a visor. I have a baseball cap but this is from Mauritia. So that is my outfit. Okay, I'm gonna do my quick divo and then head out. As always, 40 day prayer challenge. It's my favorite one ever. Also, I'm having one of these for breakfast. They are seriously just the best. I really like the vanilla bean flavor. So good. Alright guys. I said you have to say hello to the vlog. So he goes, hello. Um, oh my gosh, you're Sony dogs. My dad copied me and got Lucy. Come here. Coco's not being nice. This is Lucy. I don't know why that smudged. Okay, so we're gonna go to walk now. It's been real. Aw, I feel like it's so cute. I just don't know what's going on because I've had caffeine. So part of me is like maybe it's like a hormonal thing. I don't really know what's going on honestly. I stopped at La La Land. It's like totally safe. You sit outside, social distancing. It's been open for like a month probably now. Um, so yeah, I don't think we're on the list. And then I think I'm going tomorrow again with Liz to first Mondays, which is like a flea market type thing in Canada. It's like an hour away. So we're gonna go to that. Um, and then I think she's probably coming over today. Before anyone comments, obviously I'm being so over the top safe about social distancing. I really don't leave my house. That was like really something. But we're like taking the right precautions, wearing masks, all that stuff. And you can like social distance and it's totally fine here. It's been lifted for a while. We're already at 53% capacity. Um, I haven't even gone to like a restaurant. I don't really want to go to a restaurant still. I have another package from, oh my gosh. Not real. Okay, so she gave me another matching set. So this goes with one of my hoodies that's upstairs. My crew next, I mean so cute. I know she's like really popping off this quarantine, but she literally made me a matching like outfit for Coco. I'm literally dying. Coco, she's sleeping somewhere. She is really had the best time ever today, but she's tired now. But guys, that is so funny. Oh my gosh. Thank you so much. That was just, oh my gosh. That was seriously so cute. So Brooke will be proud of me. I finally got an iPhone tripod because all of my photos are mirror pictures, which I'm fine with. I actually prefer mirror pictures. Um, but I felt like I needed to spice it up because my Instagram is literally all mirror pictures right now, which isn't good. So anyways, I have to figure out this headache. I need to fight. I need to eat. Um, and I'm going to actually figure out what I'm making in order groceries. That's what I'm going to do. Um, I don't know if I'm going to go lay out today. I just, maybe I'll do some editing. I just love my slow weekends, where I don't have like 15 things I have to go to. It's so nice. Liz is coming over in a few hours. Um, my headache is like killing me. So I'm going to drink a Sprite because like maybe that'll help. Um, and simmer with crackers and just lay down and the jar can watch, um, some shows, update on the den. I'm going to have to switch. There's still more to be done guys. There's always going to be more to be done. I haven't even like changed yet because my head just hurts so bad. Like, oh my gosh. I don't know why I keep having these really bad headaches, but Hi Coco. Okay guys, my stomach really hurts. I have a really bad headache and it's just not the best. Anyways, it's almost three of my groceries got delivered. I was just downstairs trying to nap and it like didn't really work. I watched a few shows, some Jessica Simpson YouTube videos, the usual. Anyways, oh my gosh, my stomach hurts. The groceries have been delivered. I'm going to show you guys the grocery haul since you guys seem to really like those. This week I am making this like you guys will see tonight. What I'm making and then I'm also going to make like chicken lettuce wraps and then it's got a few other things. I have home chef coming on Monday. So most of my meals I'm going to make from that. I got a lot of chicken broth, um, because I'm just going to making a lot of stuff in my crock pot, especially lately and it always calls for that and I used it all. More balsamic vinegar up for my salads. I go through that pretty quickly so I got two. And then I got soy sauce, seasoned rice vinegar and another sauce for the chicken lettuce wraps. Dave's bread, the best bread ever. Some mushrooms what I'm making today. Blueberries, more mushroom, the big dish. Then this is for the lettuce wraps, butter lettuce, more olive oil, green onions, lots of ginger, more onions because all I do is cook with onions. You literally go through so many onions.  That will be for tonight.  So that is everything I picked up for right now. My mom is texting me, um, but I'm going to get started I think. What is it? Three? Yeah, I'm going to get started on, um, my meal today because it takes about four hours. It's a crock pot thing so, um, yeah, I'm going to go ahead and prep that. I'm actually not going to show this in a vlog. I'm actually going to show this in a vlog and not on like a TikTok or anything because I just like don't feel good enough to like do anything more than that. Okay guys, I am making braised beef with red wine. Liz is coming over. She brought this over, um, like last week and hated it. So it's actually perfect because it's what I need in this recipe. Mushrooms, beef, tomato paste, carrots, all this stuff. I don't know if that's everything I feel like it is, but I'm going to make it in my, um, crock pot. So I just browned the meat, um, I'm currently sautéing the vegetables. I'm going to add in garlic in a second and then we add in tomato paste. I'll kind of just take you guys with me. Um, this isn't like technically a cookie with kimchi, but just a little behind the scenes. Just added in some tomato paste and then I'm going to add some flour so it can be thicker. So I just added in what I last vlogged, um, but it is slow cooking. I don't know why there are timers even on. I need to fix that. Um, it's right above the wine line and then I just coated it and put time like literally, put seasoning like literally all over it. Um, anyways, you have to let this sit for about like two and a half hours. I want to say maybe three hours. I'm going to put a timer on because I don't know why it's even doing that. But okay, I'm going to do two hours and 30 minutes. So anyways, it's going to break down and just be a beautiful. We're braising this amazing. All right guys, we're on the roof. Literally my self-tanner has stained my swimsuit. So perfect. We're taking photos. Um, that's pretty much it. I'm really hungry. Food will be done in like 10 minutes. Okay guys, this has been done. It looks amazing, but it's just, um, it smells so good. We just are craving pizza. I actually made this for my family anyway. So we're getting pizza, pasta, garlic knots, all of it because it just sounds so good. Um, which is fine because this was for my family, technically. So I just didn't really care about eating it tonight.  Okay, we got dinner. Very exciting. Here we go again with our scary movies. Okay, so it is currently nine. Liz just left. I actually started watching Defending Jacob on Apple TV. I think I might even start, I don't want to start from the very beginning because I'm already one episode in. It's really good so far. My mom texted me and was like, you have to watch this. So that's what we've been watching. This may be the first night in so long that I haven't read before bed. I literally have read every single night for like the past two months. Oh, nope, still not working before bed, but I kind of think I'm just going to fall asleep here. It's very interesting, very new for me. Good morning everyone. It is Sunday morning. I'm about to get ready to head off to the flea market. Good morning everyone. I just put my hair through the Dyson Airwrap. Yeah, honestly, I'm just in such a like mood. I know I've already mentioned it all and it's not that it's ever like, you know, not worth mentioning. But it's just like, you know, I think also people come to YouTube as like an escape as well. With that being said, it doesn't need to be like ignorant. Like you need to be informed and educated in doing things. But just for some light hearty content sometimes. So I'm still trying to do that while also being like, this is a very, very big problem.  The graphics that are going by were literally my best friend Dom and it's like incredible. But we've been FaceTiming like every day and we're going to do a podcast episode as well. A lot of you guys have been requesting that. But also, I'm just like so mad, I know. Also, I just think, I don't know, I just think it's so important. So we were just FaceTiming about all of that and she was calling me because her account was getting blocked and all this stuff. And it's just like, it's crazy. And I know there's probably people which blows my mind that watch this video and still like don't. And I know there's probably people who are watching this video who like still don't believe racism exists. Somehow, even after all of this stuff. And I just like really encourage you to like, I don't know at this point. Like I literally don't know what to tell you. Like, I mean, it's a thing. It like really does exist. Like the people who are literally dying.  Like I'm white and privileged. Like it's nothing in comparison to what like anyone else is facing. But I just want to say like, I hear you guys and with you guys, with that being said, Domina are going to do an entire podcast episode. Just on like the topic. And I want it to be very like informational for you guys and for myself. Like I'm constantly trying to learn and I'm not perfect in this. Like I'm doing my absolute best with like using a platform and speaking out. And just being educated, like knowing what I'm talking about. So sorry, I just like felt so weird not talking about it because that's been like my morning.  Like I cannot imagine what it's like living this way and feeling this and like every day. Like I just like, I'm so sorry.  Like it's just so big. Like so big. So with that being said, I'm going to try to get a better video.  I'm talking about this, but it's okay. Okay, so guys, I'm just like, I'm so sorry. Like so sorry. Also if there are like more ways that you guys think I can do more things or like, you know, I'm literally always down to hear like constructive criticism. I'm not even like, I'm not offended. I think it's so interesting how people are like, oh my gosh, I mean bullied for not speaking out about it. It's like, you're somehow making this about yourself. Like obviously it's like you post anything unrelated and people are like attacking you right now and it's like, it's literally not about you. Like if people do that to me, I'm like, you're hurting and I see that and I hear that and I know that I'm not going to make this about me. Like, and people are still doing that somehow. And I'm just like, what? Anyways, so if there's anything that you guys think I can do more. Any ideas you guys have, like anything about that sort. I'm literally more than open to hearing it. Anything I'm doing wrong, whatever. Just like, please let me know.  I did Super Goop. Obviously my glow screen underneath. Shall I tell everyone to go? It's pretty similar to like what I have been doing. I'm trying to like really bronze out my face right now. Even though I like don't really need to. I used, okay, so the self-tander I used was the one that Julia Haven always talked about and pretty little thing. And I loved it. I will say like a lot of it washed off. So it looks a lot more like natural than I did yesterday. Like yesterday was a lot, but I was still like with it. You know what I mean? Blush. Blush is really, if you don't wear blush, you need to because it really changes the game. I kind of just like right now I put it around my face too to have more color. I guess my favorite makeup right now really is Charlotte Tilbury. Like it's what I used 24-7. Back to this palette. I think this bronzer is the best bronzer ever too. I'm just like, I love the highlight. Also, this is like even more controversial.  Like pray for empathy because I don't understand like how you can be Christian and like hate people. Like it doesn't, it just doesn't work like that. Like at all. So I know I'm back to this, but this is literally like the angriest like makeup routine ever, but I'm so sorry. But like in all seriousness, I'm really trying to like, okay, I'm trying to like not come at people, but like especially I think any human being regardless of what you believe, but especially Christians need to be like using their voice right now and like standing up and talking about everything that's going on. And it's just like, yeah.  Like especially you're supposed to be like fighting for people and like unity. Like in order to get unity, like people need to be heard. It's not like, oh, like this is just gonna be peaceful immediately. Like no, you need to be fighting for people. And yeah, so I love all of you guys. I just like really, really care about this and I really, really just, I don't know. Nothing I say will ever be enough. Ever. I don't think anyone says will be enough, but yeah. I'm putting on, this is literally feels like the weirdest video I've ever filmed. I'm putting on mascara. I'm about to step my face and then I'm gonna change. So Liz and I are going to the flea market. Yesterday when we were laying out, we talked to Dom for a while too. And I've just been learning so much from her and her Instagrams as well. My best friend Maddie Chop, you guys know her, growing up and still like now to this day, has always like informed me and just been like very proactive and like Black Lives Matter and just like very educational.  So because of her growing up, honestly, I really solely credited to her because I grew up in a very like white privileged, you know, area. Like that's, I feel like that's most of us.  I feel like because of her, I grew up with a lot more awareness to the matter. Like I've always, always been like racism exists. But I think she really helped open my eyes at like a younger age to a lot of what was going on and she really educated herself and she really took the time and like I'm really grateful for her, especially in times like this because I'm so glad that I was just made aware of younger, you know, like I'm really grateful for that. I also go over this with this Wonder Globe, just so everyone's aware. So I'm really grateful, but I just think like now is the time to educate ourselves. Donate.  I have been donating. I just feel weird. It's this like weird thing because if you post that you donated it makes it look like you're trying to like amplify yourself. But if you don't post that you're donating, people don't think that you did it. So I chose to not post that I donated, which maybe was the wrong thing. Like please inform me on like what I should be doing. I have shared where to donate though. So like I don't know if, you know what I mean? Like am I supposed to put, I don't know. I don't, like the last thing I want to do is like amplify myself in this. So I'm going to calm down and then we'll go on. It just feels weird to like go on with the day. But I will have places to donate, places, petitions to sign, all that stuff like down below. Actually, especially I will have Dom's Instagram page because I would rather give you one link to where you can see everything at once. So I'm going to give you guys Dom's Instagram link because she has like where you can donate, where you can do positions, all that stuff as well as like more content on like how to help them be an ally. Here's my outfit. Ignore the pillows. The shams are not here yet. Um, so I actually think maybe I have lost we because I remember these jeans being like more snug but like I, I don't know. I mean, I don't know what's happening. Maybe these are just, maybe these just haven't been washed. I don't know. I forgot even have these shoes, honestly. They're girlfriend denim just because I haven't been wearing jeans. I have a white tink on my current fave purse Air Force ones. It's just like a very chill look. I have these dresses that I'm bringing in case it's like too high and I just want to change. I don't know. We're being extra, but we're driving like an hour away. So I've really packed up. Um, that has like sunscreen and stuff in it. And then also a bunch of like just athletic wear that like I'm not wearing that lose can have because she needs some and I need to get rid of some. So there we are. I'm going to have a mush for breakfast, fill up a water bottle and we're off. Wow. This is literally so Texas and we're listening to country music. There's an eye. Okay guys, this is like actually not okay. We, oh yeah. Okay. It's so I facetimed her and she was like coming and I was waiting and then we were wearing the exact same outfit. I mean, it's not like it's a revolutionary outfit in our defense. So it's fine. We're hoping they definitely don't take them out. No, no, no. But what are we going to do? Is there an ATM close? What if, okay. Okay. What if we just floored our way through it? I think we're just going to try to flirt guys. I've done it before. I'm not a good flirt except for if I'm trying to get something that I want. Oh no. Oh my God. Wait, he's literally right here. Okay. This is so awkward. Hold on one second. Okay. Can I take cash, huh? Yes. Okay. Where's the closest ATM? I'm not sure. There's going to be one in there. But if you want to just get cash while you're in there, pay me on the way out. Okay. I have three dollars but I'm honest with you. Okay. Perfect. Thank you. Thank you very much. We love him. I didn't even try but he was just so nice. He was like no effort at all. He was like sure. Yeah.  So we just drove, it was only like a 50 minute drive. We listened to country music. It's just really been a good vibe. I'm excited. I definitely am getting hungry now that you say that though. I'm getting really starving right now. Yeah. But it's not even. It's like not even 11 I don't think. Yeah. Our original plan was to get here even earlier. And like it just didn't happen. Finally we got somewhere and she's like no we're going to eventually get there. Like at some point. So oh my gosh. Don't worry guys. We will be taking proper precautions. We will be wearing masks all that stuff. Cold pork sandwiches with soup baby raised barbecue sauce guys. Little booth that Liz found for two dollars. It's going to be great decor. Putting it in the car but it looks amazing and a strawberry lemonade. Guys seriously I mean Liz found both of these but my best buys legit ever. Like two dollars 25 dollars. I've never seen something more me in my entire life. Very successful. I mean yeah. Okay. Well we were trying to find a bar card for it. Look there's just nothing here for bar cards unfortunately but we're still on the hunt. Back from the flea market. I also got a new pair of sunglasses because I broke my Amazon ones yesterday. I don't know where I put them. Where are they? Right here. Okay. They were like ten dollars. I love getting little sunglasses at the flea market. I'm going to go shower obviously. Do all that stuff now that I've been in public. But literally so obsessed with this hat. This is the best thing I've ever purchased as well as this. Like this was two dollars. Oh my gosh guys. I just love it. Hi guys. Okay. I'm really really tired and just like sad. So Dallas actually just said curfew. You would not be able to access these areas. So I guess for several days we have a curfew at 7pm. I want to like educate myself even more so I think I'm going to watch some documentaries and things too. It's only like 3pm or it's 4pm now. I have leftovers from last night so I might just make those. And I might bring my laptop down and start editing or something. I don't really know. I want to go to the den but it's just like so dark that I just fall asleep. Okay so first Monday is just fun. I just fell so weird filming. First Monday is just fun. We got some things. I already showed you guys. I just am like going on and on. Also the Maui. Oh my gosh. My ringtone guys. The Maui Bay Browning Lotion is incredible. I really really love this stuff. I actually think I got tan. I saw a tan line when I was just showering and changing and I was like wait a second because I used it yesterday but I had self-tanner on so I just couldn't tell. I also have bug bites all over my legs and there's always mosquitoes in Texas so please tell me what to do. I don't want to wear bugs. I mean obviously I know bug spray but like is there something better than bug spray because there's literally bites all over my legs and it's just not cute and it hurts so yeah. Wow. I'm actually going to wear this. Don't worry. It really truly is just decor. So here we go. Great guys. So guys I have pizza downstairs and I could heat that up but I mean I need to do that actually. I decided I'm actually, well it looks weird without the other one but I'm actually going to hang out in my room. I have pizza, garlic knots. I'm going to turn my TV on, watch some documentaries to kind of educate myself and get some stuff together. Maybe even more resources I think for, I love you so much because I don't feel like I've shared enough on bad Instagram so I'm going to try to get some like books and stuff to read. Going to end off the vlog by just doing like skincare I guess. I don't know guys. I'm going to use pretty much the last of my Sonia de Car face cleanser which is like my favorite right now. I need to tan my face. My face is always so red after I like wash it. This is what I shared in my last favorites video and you guys are so right, duh. The reason that my soul tanner isn't working as much is because I started using retinol but the thing is like I haven't really used my retinol like routinely. Honestly like very well lately. So I'm also like maybe that isn't why. It is 5 p.m. and I'm just going to watch some documentaries and just educate myself tonight. So I'm starting off with a 13th. Hope you guys enjoyed this weekend of my life vlog. I love you guys so much. I will have places linked down below that you can donate to. Also of course if I'm telling you guys to do something I'm doing the same I would never tell you guys to do something that I'm not going to do. So love you guys so much and I will see you soon. Bye.