So good afternoon. So now we have the second lecture on algebraic topology. So we had introduced the singular homology. So that means we had to introduce the chain complexes. So that means one has some maps, Cn plus 1 goes with D to Cn, goes with D again to Cn minus 1, and so on. And then we had to define the homology of such a chain complex. So we had h, say n of C star, would be the n cycles divided by the n boundaries, where the Cn of C is the kernel of D from Cn to Cn plus 1, n minus 1. And the Dn is the image. And to make this a chain complex, we need that D composed with D is always 0, which decides we need that this image will always be contained in the kernel. This definition makes sense. And for the singular homology, we had to define this chain complex by making Cn of x, where x is the topological space, to be a formal sum, some sigma, a sigma times sigma. So this is a finite sum. A sigma is an integer. So finally, many of the a sigma are non-zero. And the sigma is a singular n simplex, so a map from the standard n simplex to x, a continuous map. And then we had to find some differential on that. And the homology is the homology of that.  And for this, we had first looked at it more abstractly again in this context of chain complexes. So if C star D and D star D are chain complexes, then we had defined a chain map, f star from C star to D star is a map at all levels, so is given by, say, fn from Cn to Dn for all n, such that it commutes with a differential. So if we have this map, so this is fn, this is fn minus 1, this commutes. So if you go this way, this way, you get the same as if you go the other way around. And we had seen that if we have such a chain map, then we get an induced map on homology. So if f star from C star to D star is a chain map, then we have a map which I call also f star from Hn of C star to Hn of D star is induced map, which is given by, if I take the class, the homology of the cycle alpha in DnC, then this is just mapped to the class of f star, fn alpha, if you want. OK, this is where we were. And now we want to apply this to show that homology is functorial, so compatible with continuous maps. So this is the following result. So well, first, we want to define the push forward map to let f from x to y be a continuous map. Well, there's an obvious way how you can make out of a singular simplex in x, a singular simplex in y, just by composing with f. So if sigma from delta n to x is an n simplex in x, then f, I mean, I don't know whether it's definition, f composed with sigma. Obviously, from delta n to y is an n simplex in y. And obviously, we can use this to define the push forward map on cycles, on the singular chain complex. But just so we define fn from c star of x to c star of y in the obvious way. Namely, if we have a cycle here, this is of the form sum sigma, where sigma runs through the n simplices, the singular n simplices in x, a sigma, which is an integer times sigma. Well, if you want to apply fn to it, the only thing we can do is we apply it in some sense to sigma, and the way we apply to sigma is by composing r sigma times f composed with sigma. And this, obviously, is a map, and it's straightforward to see. So if you want an easy exercise, why it's not even an exercise, it's trivial, if we have this f star given by the fn is a chain map, c star of x to c star of y. Because obviously, it's compatible with taking difference, if you remember the definition of d. So thus, we have a push forward homomorphism. So this is a chain map. So thus, there is a map, a group homomorphism, fn, f star, from the nth homology of x to the nth homology of y, therefore, the continuous map, which is given precisely as we had the induced map on homology for chain homomorphism, because we apply. Namely, if you have a cycle alpha, this is mapped to fn of alpha. So the class of the cycle alpha is mapped to the class of fn of alpha. So if you want to write explicitly, the class of some a sigma times sigma is mapped to the class of some sigma a sigma times f composed with sigma. And this f star, and this is true for all n, f star is called the push forward. So thus, we have now associated to every topological space sequence of groups, and to each continuous map between topological spaces a homomorphism of these groups. So we somehow see that things start translating into algebra. So we can see that also by definition. If f from x to y and g from y to z are continuous maps, then if I take the push forward by the composition, this is the composition of the push forwards. And this is completely obvious from the definition, because this is just given by composing with the map. So if you compose with the composition, you compose twice. So this is real, and it's also obvious. If we take the identity on x, with this 1x and just the map, it sends every point of x to itself. So let 1x from x to x be the identity on x. Map on x. Then clearly, the push forward by the identity is the identity. Now this is also if you compose with the identity, you get the same thing. So this is usually what one says that if one knows the language, that the homology is a functor from topological spaces from the category of topological spaces to the category of groups, being group, whatever. But we will not use this language, but we can use a simple corollary that homology is invariant under homeomorphism. So if f from x to y is a homeomorphism, so a continuous objective map, which has a continuous inverse, then it follows that the push forward by f from hn of x to hn of x is an isomorphism of groups for all n. And this is obvious from what I wrote here. I spell it out.  So we have the identity on the enthromology of x is, as we know, as I just wrote there, the push forward by the identity on the enthromology. Now we have f of f to the minus 1, but if we first apply f and f to the minus 1, we get the identity. So this is the same as the push forward by f to the minus 1 imposed with f, which is f to the minus 1 star composed, f star. So what this says is that we find that this composition is the identity on enthromology. And obviously, the other way around, we also get that the identity on the homology of y on the homology of y will be f star composed with f to the minus 1 star. So we find that indeed f star is an isomorphism with inverse, the push forward by the inverse. So this is completely standard. Now we want to finally compute the homology in one example. And precisely in the simplest example that exists, which is one point. So it's kind of comfortable. I mean, until now we have this rather complicated definition, but we haven't computed any case at all. So we want to compute the most trivial thing we can imagine, which is a point, for example. So let the point be the space, so point be the space consisting of a single point. It's a set with one element. Obviously, this has a unique topology, in which this point is both open and closed. So now we want to compute the homology of this wonderfully complicated space. So first, there are not very many maps from any space to a point, namely always one, which maps everything to that point. So whichever space you have, the only map from that point is the map which sends every point of the other space to this given point. So for all n, there is a unique map, which I maybe call sigma n from delta n to this space. Maybe I call this b. So which sends, namely, the constant map. Any x here is sent to this point. For all n, this map I call sigma n. It's just a constant map. So from this, we can already compute what the chain complex is, what the singular change is. So we get cn of x is the set of all linear combinations of all possible maps from sigma n to point. So this just means all multiples of this one map. So this is just z times sigma n. So this is cn of x is isomorphic to z. The elements of it are a times sigma n, where a is in z. And so for all n, and what is the differential? So if I take d of sigma n, then we have to recall the definition of e, which I did not at the beginning of the lecture. But how did it go? This is sum i equals 0 to n minus 1 to the i sigma n composed with this map e0. We leave out i, where this is a map from delta n minus 1 to delta n, which sends the linear combination of the e0 to en minus 1 for the corresponding linear combination of the 0 to en, where ei does not figure it. And what is this? Well, this composition, whatever it is, is always sigma n minus 1, because this is the only map that exists from delta n minus 1, so x. So this is just an alternating sum of sigma n minus 1. And so this will be depending on whether this number's even odd. Sometimes plus and minus cancels each other out. Sometimes they don't. So this will be 0 if n is odd. Or if n is equal to 0, because this was just trying to be 0, in case n is equal to minus n. Yeah, it's just not there if n is equal to 0. And so we don't have anything at all. And the other case, it is just if you look it up, sigma n minus 1 if n is even and different from 0, or maybe n bigger than 0. And I could also say, in principle, OK. So here you can just see if it's even there, one remains, all cancels each other out of one. Well, so what does it mean? Well, so if you look bn of point, so I claim that this means, if I take the boundary, the singular boundaries, this is equal to n bigger than 0. Because whenever, so this map will be either in the even case it will be subjective, but then the next map is 0 because all it is all the other around. So in every case, the boundaries are precisely the cycles. If you just look at what this gives. There's always one 0 map, and then basically the identity from z to z alternated. And in the case n equal to 0, we actually find that we start, so c0 is z, but the map is 0. So the zn is z times sigma 0, but it's not the boundary. So we have that, but p0 of point is equal to 0, z0 point is equal to z times sigma 0, so that's it for us. So the result, therefore, is that the homology of a point is equal to 0 if n is different from 0 and z times sigma 0, or just z, if n is equal to 0. So we have succeeded in computing the homology of this very central space. So you just have to look what this means and see the obvious. No, just can you, what does h0 be 0? That doesn't make sense. I mean, what can you just say, maybe you formulate in a different way so that I can understand. Yeah. What does it measure? OK. So the question is, h0 is the 0 homology. What does it measure? And you are very lucky because that's precisely the next result that we are doing because the h0 is always z to the number of path components of the space. So in this particular case, a point has one path component, and so h0, therefore, is z. And that is precisely what we do as a next step. So this is, so I'm kind of lucky. So anyway, so the answer to the question is the following. So first, so this is a bit more general. So if xk, k is the set, so let xk, k be the set of path components of a topological space x. So we say that if you have a topological space, we say that two points are in the same path component if you can connect them by a path. So map from an interval x, and there are different path components if you cannot, like here. Then the first statement is that the homology of x for all x is just the direct sum of all the homologies of the path component. OK, so just have an, if you have a kind of obvious so that elements in the homology of x are the sum of elements in the path components. If you have k, if you have the two path components, x1 and x2, then the nth homology of x is the nth homology of x1 plus the nth homology of x2. But in principle, it also holds if you have infinite many components. But you don't have to worry about that too much. And the second statement is that if x is path connected, then the 0th homology is just z, isomorphic to z. So in particular answering your question, so it follows that if x has, say, k path components, then h0 of x is isomorphic to z to the k. So we can really say that the 0th homology measures precisely the path components, the path components. It just measures how far x is away from being path connected. And so the way how one finds out anything about connectedness properties of the path connectedness of space is by homology is decided by computing the 0th homology. OK, so this has these two parts. So the first one is in some sense trivial. I only have to see whether I can convince you of it. So proof. Well, we have this n-simplex, delta n, which is a simplex in Rn. So certainly it is path connected. It's actually a contractable space. So if sigma from delta n to x is a singular n-simplex, then it's the image of the path connected space by a continuous map. So then the image is path connected. So it's path connected, and therefore it lies in one connected component of x. And so furthermore, if we take d sigma, then all the, then this is a linear combination of n minus 1 simplices of singular n minus 1 simplices which lie in the same path component. As is so where an image where sigma composed of n is a singular simplex in the same path component of x. So if then we write x of xk, where the xk are the path components, then we see that every singular simplex lies in one of the xk. So if you have a chain, it's just a direct sum of change in each of the xk's. And the differential just sends something which lies in xk to something that lies in xk. So we get that c star. So it follows that cn of x is just a direct sum over all k of the cn of xk. The d reflects this summation with something like the cn of xk, then d of it lies in cn of xk. So it follows that also this is true for cn and dn. That would be not so n of xk. And then as the homology is the quotient, we get that the homology of x is equal to the direct sum of the homologies of the path components. So maybe I did it fast, but it's really trivial. After all, a singular change is just a formal sum of simplices. And each of the simplices has to lie in one of the path components. And everything is just a direct sum. So now we come to the second part, which is more interesting. So we want to see that the zero homology is z. So we have to somehow construct an isomorphism. And so how do we do that? So for any point p in x, I write by abuse of notation if you can handle it, if not then I could change it. And I write p also for the map from delta n to x, which sends all points here to the same point p, so the constant map. So I just call the constant map to send everything to p, also just p. So this should not lead to confusion, but if it does, then OK. So if you have a point, say x in x, and now the point is called x, well anyway. So let's say for point p in x. So let's choose a point, say x0 in x. Now we have our x, maybe we have any point p here at x0. Then x is path connected, there will always be a path connecting them. So for p in x, let's say sigma p from delta 1, which is nothing else according to our definition in just the interval 0, 1. So x be a path, so we choose a path from x0 to p. So this path here is sigma p. Then if you remember the definition of the differential, the differential of this thing is p minus x0. So the constant map p minus the constant map x0. So then by definition, if you look what the differential means, in this case where you just have one, there's just two points, the beginning point and the end point, the differential is the end point minus the beginning point. The definition is d from sigma p is equal to p minus x0. So for every path, we get that the differential of the path of this difference, this formal sum. So now we define a homomorphism epsilon from the 0 chains of x to z, which is if we have a 0 chain, this is a formal sum, I may be calling it p. So take the points a p times p. This p is the singular, this map. Ah, yeah, I actually don't need that. It's a very constant map. It's just the map that sends the point to p. So that's the way I want it. I didn't want it for all n, but just for n equals 0. Sorry for that. So maybe I can repeat. So we defined the p to be the map which sends the one point that this thing has, which is the point 0, to p. So somehow it's quite reasonable to identify the map which sends one point, the one point of this thing to one given point with this point, because that's precisely information that it contains. And then so this was our p. And now given a point, we find that we can make a path to any given point as the differential of the path need as the one chain is precisely the different point minus the kind of base point we chose. Now we define the thermomorphism. So if we have this, we just take the sum of the coefficients. Now by definition, the chain is a finite subsum. So only 5 to many of the a p, which are integer and non-zero. So this makes sense. The finite sum of things. So now we want to prove that the homology is isomorphic to z. So we know that every element in c0 is also a cycle, because the differential for the next level is by definition 0, because below 0 everything is 0. This is also the same as c0 of x. So then the claim is that the kernel of epsilon is equal to p0 of x. And note that this proves part 2. It proves 2, because epsilon is obviously subjective. No, we can write down anything. We can get whatever we want. We just take one point and take any coefficient. Then we get that coefficient. So the map is certainly subjective. And if we have a map from v0 of x to z, which is subjective and this kernel, if I see this, certainly is an isomorphism, which is an isomorphism of z of the quotient of v0 by v0. z0 of x, I'll just divide by v0 of x with the map induced by epsilon to z. So this is whatever, the homomorphism theorem. And after all, this thing is h0 of x. OK, so we just have to prove this claim. Well, if you have to prove that two sets are equal, you have to prove both inclusions. So that's what you learn. So let's try both inclusions. So let's take, say, an element equal to some p, a p times p, p an element in the kernel of epsilon. So then we have to show that it is a boundary. That is d of something. Well, that's quite simple because we can write it like that. So this is certainly alpha. But we can also, you know, the sum of all the a p is 0. That's precisely what means it's in the kernel of epsilon. So we can add 0 for this thing. It's still the same thing. So this thing is actually 0. But now we see if we look at this thing, this is just the differential of this. So this is equal to the sum over all p, a p times the differential of sigma p, which is the same as the differential of sum a p sigma p. So we see this is indeed a boundary. And the other direction is even slightly easier. I think so if, say, sigma from, again, delta 1 to x is a singular one syntax, which is the same as a path, then what? Well, obviously, we have that by what we wrote here, that the differential of sigma is sigma of 1 minus sigma of 0. This is precisely what it says, which is the path from sigma of 1 to sigma of 0's differential is this. So in particular, so maybe to be sure, if I write something like this in the other notation, you could, like here, you could also write this as plus minus this. So this is what it means as a linear combination. Also here, I could say this is this plus minus 1 times this. So that's what this notation means, like it means in high school mathematics. So in particular, if we take epsilon of d sigma, then this is the sum of the coefficients here. So this is 1, it's minus 1, so this is 1 minus 1 equal to 0. So if you have any cycle, so if alpha is equal to sum sigma a sigma times sigma is a one cycle, is a one chain, then. And we take the differential, then so epsilon of d alpha is equal to epsilon of sum sigma a sigma times d sigma. And epsilon is also just linear, so it goes in here. So this is sum sigma a sigma times epsilon of d sigma. And we know that for every one simplex, epsilon of d sigma is equal to 0, so this is just the sum of 0. So we see this includes. So we have that indeed, v0 of x is contained in the kernel of epsilon. And so this is for position. So now, so much about this. So we have at least found out what the 0 homology computes. And we have been able to compute the homology of a point. So now we want to come to the first theorem in this lecture. So the first time one really has to make some effort. So until now, I mean, I expect you have to have to make quite a lot of effort to, I mean, because there are so many definitions that one has to deal with. There are lots of definitions. It's difficult to keep them in one's head. But the things that we do with the definitions are completely trivial. It's maybe difficult to notice because there are so many, everything is new. But now, we want to do something where one actually needs an idea. And so this is the homotopy invariance theorem. So it might be that the proof is actually not more difficult to follow than the other proofs. But these, if one gives you all the definitions and you understand them and tells you what the result should be, you can prove it as an exercise. But this one is not an exercise. So the homotopy invariance theorem. So we want to show that homology groups are homotopy invariance. So what does it mean? So first, I recall to you about homotopy. I mean, you have had it very recently, but I need to also fix the notation of the definition. So I write now also I for the interval 0, 1. So let f and g be continuous maps from between two topological spaces. So a homotopy f from, I write it just like this, from f to g is a continuous map from, which we want to wonder, x times i to y, such that if I restrict to the part of i, where if one is 0, I get f. If I restrict to the point here being 1, I get g. So I write it in a slightly different way because I want to use this notation in the proof. So such that if I take the map i0 from x to x times i, which sends x to x, 0, and i1, corresponding thing on the other side, if I have these two maps. So for these maps, we have that f is equal to large f composed with i0, just means the same thing I restrict to x times 0. And g is equal to f composed with i1. So this is a homotopy, and you have learned that before. So if we have a homotopy, we call the two maps homotopy. So if there is a homotopy, f from f to g, we call f and g homotopy. As we're at it, I introduce a few other words that you already know. So to say what it means to be. Homotopy equivalent, and then as a special case, contractable. I expect we have learned all this. I don't know who taught with Zimmermann or Dillegas, but anyway, just one notation on the table. So two topological spaces, x and y, are called homotopy equivalent. And if there are maps which are inverse to each other after a homotopy, so if there are continuous maps, I hope that's correct, f from x to y, g from y to x, such that if I first go from x to y and then go back, this is homotopic to the identity on x. And if I do it the other way around, it's homotopic to the identity on y. And I call if x is homotopy equivalent to a point, I say it's contractable. So to the space which I call point, a point, we call x contractable. There's somehow a homotopy which contracts the whole of x to a point. So now, we want to prove this theorem that if two maps are homotopic, then they induce the same map on homology. We know that if we have a continuous map between topological spaces, it induces the homomorphism on the homology. And now the statement is that which map the induce depends only on the homotopy class of the map. So this is the US data theorem. If x and y are no, so let f and g from x to y be homotopic, then if I take the push forward map by f, then it is equal to the push forward map for g, which has a homomorphism from hn of x to hn of y. So homotopic maps give the same map on homology. You have some easy. So in some sense, it says that the homology is not such a super fine invariant. So it only sees, for instance, it only can distinguish the homology groups up to homotopy equivalents. So if two topological spaces are homotopy equivalent, they will have the same homology, where that comes in a moment. So that tells us that there's one thing that they are not. They don't contain all the information about our spaces. On the other hand, it maybe tells us that they are somewhat easier to compute because they only depend on this kind of data. So corollary, if x and y are homotopy equivalent, then the nth homology of x is isomorphic to the nth homology of y for all n. What? Can we use this film without proof? What does it mean? I mean, we use it without proof. No, I first give the corollaries, and then I give the proof.  So I first give the application, and then I give the proof. So I think you can figure out yourself why this is a corollary. You have this, if it's a homotopy equivalent, so that the composition one direction is the identity and the other it's the identity, so that precisely means that if you do it for, you get the same thing for the push forward, and that's it. I can maybe make this, and if you cannot do this, then maybe you should already do it. So the second one is, which is a special case, that if x is contractable, then we have that hn of x is equal to 0, if n is different from 0, and it's z if n is equal to 0. This is because we know that this is how the homology of a point is, and if it's contractable, it's homotopy equivalent to a point, and therefore it follows from the previous corollary. So and now we want to come to the proof of this theorem. And in order to do this proof, we first, again, need to go back a little bit to the homological algebra. Maybe I start with a new thing here. So we need some small amount of homological algebra. Again, we have to talk about change complexes and so on. And so what we need now is a change homotopy definition.  This means that at each level, we have fn from tn to dn. Commuting with dn, we change homomorphisms. That means we have cn, both fn and gn, tn for n. Commuting with a d. So a chain homotopy from f to g is another sequence of maps which now do not, which go a little bit diagonally. So is a sequence, maybe I call them sn, from cn to dn plus 1, such that then we have this strange formula that somehow interpolates between, if we also use the differential, it somehow interpolates between f and g, such that if you take d composed with sn minus sn minus 1 composed with d, this is, so now I want like this, gn minus fn. So let me write this also in diagram form so that you have some. Anyway, we can see what our theorem is. So we have here, say, if you have cn minus 1, cn, if you want 1 cn plus 1, each time d. And we have here the same with the d. And now at the, so we can do the following. I'll write it here. So what is this map? So we have here the map sn. Here we have the map sn minus 1. Here we have both the maps fn and gn. And now, so this is supposed to be for all n, obviously. And so now we want that if we do, say, if we first go here, and then here, and we subtract first going here, and then here, we get the difference between these two maps. So the difference between the two maps, I mean, after all, these are homomorphisms of the Boolean group. So I can form the difference. The difference between these two maps is the difference between commuting these maps with a differential in the two different ways. So this is a chain homopop. And why would we be, why would this be of any relevance? The point is that if we have a, the claim obviously is that if you want to use this, that if you have a chain homopopy to chain maps, then they induce the same homomorphism on the homology. And then we will use this to prove our theorem to show that if we have a homotopy between two maps, then we have a chain homotopy between the induced maps. I mean, between the induced things on the chain. So let's see. So we first have to prove this little lemma, which is actually quite trivial. That's one of the features of homological algebra that most arguments are extremely simple. It's only that there are very many very simple arguments. So for position, let f star and g star be chain homotopy. So from, say, c star to d star, the chain homotopic chain maps, chain homomorphisms, then, well, if I take the map on homology, it's equal to the given map of homology. And you'll, OK, in order to prove this, we have to remember how this map was defined, the induced map. So how was the map given? So f star of the class of an n-cycle alpha, so if alpha is in zn of c star, then f star of alpha was just the class of fn alpha. That's how we have to find it. And you'll have to see that this is where you find. So first, if alpha is a cycle, then fn alpha is a cycle 2. And if one changes alpha by boundary, then also this gets changed by boundary. But this is kind of trivial and trivial. So now we can want to see that this also works here. So we know what the map is. So let, we are here to prove. So let, so alpha is here. So we have, if we take alpha in, so is a class in the nth homology of x, then we have that alpha is 0 cycle in n-cycle. So that means d of alpha is equal to 0. So now we can try to form the difference between the push forward by f, the map induced by f homologian, the map induced by g homologian. So if we take f star alpha minus g star alpha, what is it? Well, certainly we can, by definition, this is the same as putting the fn into the stain. And then it's just a homomorphism. So we can also put the sum into it. So this is the class of fn alpha minus gn alpha. And now we look at this thing. We have our chain homotopy. So we write it down in this form. This is d of Sn of alpha minus Sn minus 1 of d of alpha. So alpha is a cycle. So d alpha is 0. So the whole thing is 0. This one is 0. So this is the same as d of Sn of alpha, the class of this. But this is d of something. So it is a boundary. So the class of d of something is always 0. Because after all, the homology is precisely given by dividing by the boundary. So we see that the difference between these two is 0. So the maps are the same on homology. So now let me see. So this is the general quarter of now. Now one comes to the actual proof of the theorem. So we'll somehow start with the reduction, which is easy. And then we'll see what we can still do. And at least tell you what the setup is. So now we come to the proof of the theorem. So recall what the theorem is. That if we have two homotopic maps, f and g from x to y, then the induced map on homology is equal. And we want to prove this by eventually constructing a chain homotopy between the induced map on the singular chains. But first we want to make a reduction to a very special case. Somehow there is a very simple homotopy between two maps. And it's enough to prove it for this one homotopy. So if I take the identity of x times the interval 0, 1, this is a nice map from x and p. So this actually is a homotopy from between the two maps, i0 from x to x times i. If you remember, this was sending x to x comma 0 and between and i1. This is obvious that this is a homotopy. It's a map from this from x times 0, 1 to some space. Such that if I restrict it to x times 0, I get i0. If I restrict it to x times 1, I get i1. It's trivial. So the claim is, however, that it's enough to prove the theorem for this idiotic case. So assume we know that i0 star is equal to i1 star as maps from hn of x to hn of x times i for all i, for all n. So assume we know our theorem just for the kind of trivial homotopy between these two maps. Then I claim it holds for every homotopy because we can get every homotopy by composing this thing with a continuous map. So let f from f to g be a homotopy where f and g are continuous maps from x to y. Well, then it's not very difficult to see that. Obviously, f is just f composed with i0. g is just f composed with i1.  Nobody can doubt that. We don't really need it. So we just obtain our given the homotopy we want just by composing this kind of trivial thing with our given f. And then we can use that homology is functorial so that we have an induced map. So then it follows because of that f star. So the map on homology, induced by f, is just f star composed with i0 star. Now, as i0 star is the same as i1 star, this is the same as f star composed with i1 star. But this is g. So we just reduce to this. So thus, we note that it is enough to prove the theorem for i0 and i1. So we want to deal with this case. And now I will start doing the setup. So somehow we have to construct our chain homotopy. We somehow start with single synthesis on x. And we have to somehow make out of them construct from them synthesis on x times i. And we somehow do this by taking our simplex, making a copy of that same thing, 1 at 0, 1 at 1, and dividing this thing into simplices and doing something with it. So let me set it up. So now we come to this. So on delta n times our interval 0, 1, we have the following points. We write vi equal to ei, 0. Remember that ei were these points in the kind of corners or however we call them, the vertices of delta n. And wi equal to ei, 1. So that means if we do just the two-dimensional case, we have here, I mean, the one-dimensional case and then times i is two-dimensional. So if we have here v0, v1, w0, w1, so in this case we get such a square. But if you have another simplex, you would have n. Let's first see how it goes on. And now we define for i equals 0 to n. We want to define a singular n plus 1 simplex in delta n plus i. So we put a singular n plus 1 simplex in delta n times i as follows. Well, if you remember the notation we used to denote maps. So we take the map w0, wi, then we start with v0 for vi, then we start with wi until wn. So this, if you remember, we take this by definition is the map which sends the linear combination of v0 to n plus 1 for the n plus 1 entries. So we call it one linear combination of these. So that means, so this is a map from delta n plus 1 to delta n times i. Maybe I can just write. So if we have, this is the map which sends sum i equals 0 to n plus 1 ei, so ti times ei, where ti are some numbers which add up to 1, to by definition sum i equals 0. This cannot be i because we have an i, so I call it k. So k equals 0. So i tk vk plus sum k equal, how do I want to do it? I plus 1 to n plus 1 tk wk minus 1. So this is the map. So this is just written out what it is. And so obviously the image of this map is what I had denoted like this. So just a set of all linear combinations of these elements. And so if we look at it here, we have, for instance, in this case, we are n is equal to 1. We get two such things. We have the one map which maps the same text for this thing. And the image is this. And the other one maps the n plus 1, the two syntax for this thing. And the image is this, in the obvious question. And it's more difficult to imagine if you are in higher dimensions. But anyway, so that's how it looks. Now I want to use this to do something with it. Let me think about two minutes. So maybe I just write down the operator. And then we have to show that this gives us the event of a chain homo-copy definition. So define the so-called prism operator. So this is p from cn of x to cn plus 1 of x. So it should have been called pn, but I just write p. So I first define it just on singular synthesis. So first I define it for a simplex, sigma from delta n to x, simplex, singular simplex. How do I define it? Well, I just write down a formula. So p of sigma is equal to sum i equals 0n minus 1 to the i. So when one writes formulas, one always wants to sprinkle some signs in it so that afterwards things cancel. Now we take sigma times the identity on i. And we apply this to w0, no, v0 to vi, wi, wn. So we have these simplices, these maps from delta n plus 1 to this thing to the product. And we apply sigma times the identity to it, which gives me a map from this delta n plus 1 to x times i. So after all, that's what I wanted. And we want to, this should be our, what is not composition? Sigma equals 1. No, it might be there's a misprint in the notes. So this is not the composition, this is the product. So it's sigma on the, so this is after all we are, it's sigma on the delta n, it's the identity on. So this thing is a map, if you look at this. This is a map from delta n plus 1 to delta n times i. And now we first do this map. Now we are in delta n plus i times i. And now we apply the sigma on the delta n and the identity on the i. So this is the thing. If you look at this particular case, we apply, we take this whole thing and we apply to it sigma times the identity on i to go somewhere, to go finally to x times i. And we take here, we take it with a positive sign and here we take it with negative sign. And this has the nice effect that if you look at this thing, we have 1 plus 1 minus, and you can hope that in the middle it somehow will cancel. That's why one has this sign. And so then the claim, which will prove the theorem, is that P from i0 to, so P is a chain homotopy from, say, i0 star from c star of x to c star of x times i to i1 star. So that's the claim. And that's the thing we really have to do. But then if we have proven the claim, it follows. So then the theorem follows. Because if we have a chain homotopy between these two, this means we need the same map on homology with us. Because then i0 star is equal to i1 star on homology for all n. OK, so maybe here I stop. So now next time we will repeat the setup and then we will have to prove this claim. And with this we will have proven the theorem. OK, thank you very much.


That's what you had. So I can see why it might be difficult for a doctor to, you know, if you came in, the tests are showing negative, and you're saying that, you know, I just know that this is something that I have. But in the end, you were right. You were absolutely right. Three times, unfortunately. And what's really great is that she took power for her own health and her own life rather than just deferring to the authority of someone in a white coat and saying, oh, he knows best. And I think we all really need to learn from your example.


So it is exactly two o'clock, which is a time when we are supposed to start. I am Wissac, I am one of the co-organisers. The other one is Professor Dragos Georgiu from Romania. I am French, but I am from Toronto, Canada. And we have decided to propose that session on history and prehistory of space. So I will make a very short introduction. And after that I will give my paper. And I apologize for having scheduled my own paper as the first one. Because I am not a real archaeologist. I am a share archaeologist. That is I just read and write, but I don't do digging. And after I am finished, then we can move to real archaeologists who really confront constantly the harshness of the archaeological record. So my first few words of introduction is to ask in what sense can we say that space changes with time and thus can be considered to have a history and a prehistory. This sounds at first paradoxical since for the modern mind, space appears to be an invariant of human experience, precisely what does not change with time, but within which changes occur. We assume that the experience of a three-dimensional space is a universal category of the human mind and this is undoubtedly true in as much as all life takes place in a physical environment. However, it is likely that this abstract concept of space is a cognitive elaboration that emerged in the western human consciousness with the advent of Euclidian geometry. This is a space reduced to three dimensions without gravity and without complexity that we project upon our environment like a net or a grid in order to objectively measure distances and evaluate surfaces when we endeavour to describe and interpret the archaeological record. Another source of our modern representation of space depends on the achievements of geography. It is easier indeed to conceive this aspect of space from the point of view of history because we can compare the maps that were produced over a long period of time for various practical purposes such as navigation, exploration and administration. Although maps cannot be confused with actual territories, signal progress was made during the last few centuries to reach a close approximation in the projection of two dimensions of Earth's morphology. Today's satellite photography allows us for high definition of details but what is gained in details is lost in terms of the global representation of general patterns and relations that are preserved in our semantic memory with various biases. I should put some slides now. Can you help me please? Can you help me here? We now imagine our planet as a set of five continents and thinking of the world today calls up images that are markedly different from what this representation was a thousand years ago or more. The archaeological record provides evidence of how space was experienced, conceived and represented up to a certain temporal threshold but not beyond. When archaeologists locate, describe and artifact in space, they necessarily rely on the contemporary epistemological toolkits of geometry, geology and geography as these points of view of our proximal and distal environment stand now. No data by itself can be correctly interpreted that is cognitively represented with some degree of plausibility in the absence of information about its context. Most often this information is of a spatial nature even social information as a proximic dimension. There is a constant danger of extrapolating our own modern experience of space when we reconstruct the spatial context of the archaeological record. This issue is still more pressing when we try to describe and understand hominine and human mobility. Early migrants were bound to have a radically different relation to space since it can be assumed that their experience was not mediated by the knowledge of any form of geopolitical maps compelling them to rely on other cognitive affordances of their immediate experience. Can we infer, imagine or simulate with any kind of plausibility the prehistoric experience, conceptualization and representation of space in order to better understand the archaeological record? This is the question that our session will attempt to tentatively answer from a range of timescales and from a variety of archaeological points of view. This is a question that has haunted archaeology at least since the landmark publication of the collective volume edited by Colin Renfrew and Ezra Zubro, the ancient mind element of cognitive psychology which I think is a landmark at the beginning of the start. At the time. At the time, yes, but you have to start somewhere, raising some questions. Can we reconstruct, can we imagine the mind of Palaeolithic people hominine? It's extremely important. And I must say that there is also another important article by Ezra a prehistoric space, an archaeological perspective which was published in the Journal of World Anthropology and I don't remember the date but it is probably contemporary to this book more or less to this book. So now I will come to my own presentation and once again I apologize for scheduling me at the beginning but once again it is because I am a armchair anthropologist and there are just a few remarks and after that we will move to the real.


Welcome digital asset news or Dan for short. My name is Robin today We've got a lot of things to go over so let's just jump right in the first thing we're gonna talk about obviously is the crypto pump today So we're gonna take a look at a couple reasons why that's going on I think it's coming out two things acceptance of the Fed hikes by Jay Powell and Also institutions as they come in as a tiptoe into our market Also, we'll take a look at some positive macro trends for once and also talk about building in the bear market And then lastly we have to balance everything out So we're gonna talk about how it's not all good news though There's some things on the horizon especially some macro factors But I think if you stay with me all the way the end You'll get a little bit of a peace of mind and then lastly just so everybody knows This live stream will be reaching over into the DCA live stream with me James from invest answers and Ben from another Cryptoverse there will be a link in the description once I get it which will be hopefully the next couple minutes And that DCA live show will start actually in 45 minutes. So everybody Welcome happy Friday. Glad you stopped by first things first The crypto pump what's going on? Well as we can see quite clearly And I'm sure you all know this because you look at your portfolio way too much is Bitcoin is pumping if they're in pumping everything's pumping is great great fantastic, right? Binance coin is pumping. I guess nobody really cares about how centralized it is whatever XRP Binance one of the big winners for the day though is Kind of here's polygon just crushing it 18% and 24% and there's a couple of big news items, of course They're going to partner up with meta which used to be known as Facebook as they start to mint NFTs Also the same thing we said for Solana, although not as big of a pump for Solana And that's just what's going on. So it's interesting today that We started this pump and we keep going but if you take a look at what's going on the traditional markets NASDAQ is actually down a little bit started off Week then rally then down down it goes again same thing here. Whoops. I look it's going to go to the S&P I wanted to see the same thing twice Oh and again that one started off Same way as NASDAQ rallied came back down just kind of moving sideways So it's interesting to me right now that we in our market. We're holding up pretty strong now Does that mean that we are going to keep going up? Probably not because remember we are in crypto and there's a little bit of Volatility but I talked about this a couple days ago When Jerome Powell came out and talked about hey federate hikes are here We told you that we're gonna keep going all the geniuses were like no no They're gonna pivot or no there was some some Little dovish talk a little bit ago about this and this and everybody analyze it And I'm just like just listen to what he's saying listen to what he's saying listen to what daily is saying from The Fed Reserve the San Francisco chairwoman They're saying the same thing. We're gonna keep raising rates until we have 2% I don't know why this is so hard, but again The Fed rake Fed rate hike was expected. So I asked the question why is why did we keep having these little dips? It never fails people pay say it's priced in there's always a little turbulence and then maybe tomorrow we go up 20% Well, I was wrong about the 20% but in our market. We do seem to have a little bit of a rally So the question that is well, why is it us and not everybody else? Well comes down to a couple a couple of things first of all This was put out by ny dig or nigh dig some people call it And I just tweeted this out ago. I think black rock fidelity bny melon when the oldest banks in the united states mass mutual And again, that's I mean all combined. These are big institutions that are getting into crypto I think they figured it out. They're like, hey, guess what? bitcoin here In q3 of 2022 isn't so volatile Actually, it's doing pretty well precious metals is going great great because I own silver and gold too So i'm happy about that cash is up a little bit. Hey Good times being cash like benzo is talking about cash is king Let me take a look at uh, what's going on this quarter short-term treasuries total us bond market Long-term tax and 10 tenured treasuries us large gap us stock market. Everything's down across the board re it's Real estate investment trusts are down 13 shocker I mean as we see like uh, the amazing housing run that we just saw last year going into january and february 2022 I mean You just take a look at this this if I was an institution would make me sit up and go man Maybe it's not as volatile as what I think it is but again We can't cherry pick the data. We have to take a look at everything and there is Something to be said for institutions coming in and I want to stress this enough that there's always a story behind the story We talked about this yesterday this was from The trading volume uh from uh, it was actually from the institutional investors in q1 2022. This actually came from coinbase and This was their q1 2022 shareholder letter And 76 trading volume and coinbase came from institutional investors That's just what it is and uh, if you don't think they're here, they're here And then we talked about this again as far as bitcoin treasuries linked in the description Did you know that uh institutions own here's the number? 6.876 of the total Circulating supply of bitcoin 6.876 percent. That's a lot That's uh broken out between etfs countries public companies and private companies. So what's going on? Well, it's summed up by Frank Shaparro and he talks about as at the end of the quarter Roughly 25 of the 100 largest hedge funds in the world By reported assets under management Have chosen to on board with coinbase That's quite a bit. That's quite saying something right so frank asked the question So institutions are coming right?  I can't tell you what it is because i'm not an insider Not even a financial advisor but Frank actually a little bit digging says I just got off the phone with coinbase's head of investment research or institutional research It's important to note that a lot of these hedge funds are just testing the waters in crypto A spring is coiling and they want to be positioned to capture the markets upside When the winter thoughts, I think that's a big thing to wrap our head around to make sure that we understand that I cannot keep giving or I don't really give that much hope you But i'm not going to sit here and go it's going to happen soon. It's going to happen soon It's going to happen tomorrow. It's going to happen nice weeks and happen a month It's a long game and that's why we dolly cost average and we talked about this before as far as the stockdale paradox and stockdale paradox jim stockdale He was a naval operator A pilot and he was shot down in vietnam war and he lived in the Hanoi Hilton for I don't want to say four to seven years. It was a long time He was tortured every day very awful time, but he made it through and He they talk about what made you get through he goes I didn't expect too much hope And he said well, isn't that what it is about hope? He goes the people didn't make it Where the people they're being tortured every day? And then they would say to themselves and the other prisoners around the camp They'd say don't worry by Christmas. We're out of here But don't worry by the end by the beginning of the year new year. We're out of here Don't worry by by easter. We're out of here and it crushed their spirits So you have to understand the the gravity of the situation right now some macro events aren't that great Some things that are going on in crypto. I mean heck. We just had an issue with gala being hacked And that just dropped a little bit and of course all the defi acts It's not perfect right now, but in the long run I think we're in the right place at the right time and then that leads me to Talk about again to reiterate about institutions. Goldman Sachs Launches a crypto explainer a trend tool for institutional clients again They're not getting into it heavy But they're getting ready just like frank talked about the coil coming up. So it's called Detonomy as a nice one the tool is a collaboration between Goldman Sachs bunch of services from MSCI and crypto intelligence firm coin metrics It classifies coins and tokens based on how they're used Amos provide a standardized ways for clients to analyze the broader crypto industry Including defi and smart contract platforms And then just to note in march the bank reached on a milestone where it offered its first bitcoin back loan during the likes of Crypto from the banks such as silvergate capital corp and others. So again We see that the institutions are kind of like well We like this but we can't go on heavy because it's about wealth preservation not making a bunch of money But again, when you take a look at where the biggest upside is Would you be looking at the nasdaq and the s&p 500? Especially with the rates that are going up or maybe just maybe you take a look at something that's going on the crypto market And say wow, it's just barely over a trillion market cap Do you know that the market cap for gold is 12 trillion dollars? And then for traditional equities is over 120 trillion dollars. We're just sitting at this to me It's kind of like a no-brainer, but I could be wrong. So let me know where I'm off of my thinking in the comment section. Let's move on to some positive Macro factors. I just came across this positive positive macro factors and building on the barracks, excuse me So we'd like to talk about the cpi and the inflation numbers and things like that But this is from the bureau of labor statistics I think I linked in the description if not, I'll make sure I do but this is the import price index for selected categories and you can see that This is for excuse me. Let me change this real quick Import price. Let's do all of them import and export price indexes See what's going up and down and we can see that I mean people talk about there's a problem with With imports and we can't there's a supply side issue We can't get things and if that was really the case I think things are catching up because as far as like in import and export price index We actually peaked over here at 13 percent Same thing happened happened over here in the recession of 2008 and it started to go down Same thing is coming up over here. So maybe we go down, but that's not If you take a look at export and imports is one thing but really what I care about Is the imports? How much do we have to pay for the things that are coming in? Well, actually that's gone down too And I thought it was interesting that some of the things that are going down Like industrial supplies and materials for us to build here in the united states We need those materials and if they were they topped out a 55 percent may 2021 December 21 35 percent almost 40 percent in march And now we're down to only a measly increase of 15.4 percent But overall again, we're on that trend going down So if you want to look at some macro looking pretty good as far as imports Hey, at least some of the imports that we got coming in are actually going down and also Talking about building in the bear market You know how We've talked about different cryptos and digital assets that are trying to get into africa They're trying to say that africa is the next big play. I never really I understood it But I didn't really understand it completely and there was an article that came out It's pretty interesting. This is from the guardian. I linked this one in the description. You can read it It's very long But it blew me away because I didn't know about the population and before I go on An increase in population does not equal prosperity but I'll give you in a second so I don't even know this by the end of the century africa will be home to 40 percent of the world's population 40 percent That is a boatload. Today, africa has uh 1.4 billion people By the end of the century the um projects at africa which had less than one tenth of the world's population 1950 Less than one tenth in 1950 Will be home to almost 4 billion people or 40 percent of humanity and it goes on and talks about all the different things That's great to read. It's fun. It's great, but we don't have that much time So I just want to get to this last part Whoops not this part here this part here The best thing that could happen at west africa will be if someone could convince these countries to seriously consider the experiences of asia This is from elaine bertah senior fellow at the morone institute at new york university He's a former world bank official who specialized in urbanization bertah'd advise china about developing and remember 1950s China was the same place wasn't a big economic powerhouse But then it became such a thing and the reason that it came infrastructure builds And of course it allowed a lot of the corporations to come into its country and have sheep labor Density itself does not create prosperity Again, you will have to have lots more transportation new rails new roads that link the coastal highways to small cities With a cheaper land is now she have to have infrastructure telecommunications being one of those things However, I started to think to myself well in africa Where's the play? Where is the economic prosperity? I didn't know this But they have a lot of natural resources africa Is home to some 30 of the world's mineral reserves didn't know Eight percent of the world's natural gas. All right 12 percent of the world's oil reserves Well, we're going to need that for a while 40 of the world's gold interesting And up to 90 of its chromium and platinum Largest reserves of cobalt diamonds platinum and uranium in the world are in africa 65 percent of the world's arable land and 10 percent of the world's Internal renewable freshwater sources so every time but just natural resources now. I understand why the lot of these Projects crypto products specifically are like we need to build over there because that's the next big thing I totally understand So also Of course, we're talking about cardano. If anybody can't read between the lines. I'll help you out Cardano is just one big play and cardano has got this really great project. It's called world mobile token You know, that's already up and operating right now If you want to talk about utility, it's working Total aeronauts 152 total network consumption 24 hours 1.08 terabytes unique user 24 hour 15 000 So if we're looking at connecting the unconnected Intelli communications and mobile connectivity. That's the place and before anybody starts to talk about yeah, but what about the the elan musk option Whatever that project is calling escapes my mind right now You know much how expensive that is To do you know how cheap it is to do this? And then here's how they work it. There's this thing called aerostats There's essentially blimps they go up in the air they transfer data and you can have it for miles So for me if i'm looking at what's the next big thing The elan musk one Talk me in the comments. I forgot the name Which costs hundreds of dollars per month Or something that's going to cause pennies in the dollar Which one would you say would be the next big thing in africa? Just asking the question so those parts Also I will be selfish here. I'm very biased I own gentsukishi, but again building the bear market gentsukishi just launched their actual online metaverse Or meta world gaming experience on november 30th So congratulations to those also The cryptiverse not to be confused with ben into the cryptiverse That's another type of play is a metaverse play. They also have a video that Essentially shows what they're building how they're doing things and it's pretty cool because you actually build things in there again I own these things so i'm super biased So don't say ah rob just he's not biased. Oh rob super biased and again links in the description for this one And uh, also one of the reasons why i like gentsukishi is because you know, it's already on the app store for apple google Nintendo switch and ps4 now. It's transferring over to pretty much web 3 and metaverse place So also links description for that uh deep dive video matt would take care Of the good news, but we have to be balanced a little bit. Let's be balanced. Let's not uh get high on hopium Ah This is what we got so the housing market and just stick with me on this one trust me. It'll be okay The housing markets take a little bit of tumble again It depends on where you're at if you're talking about miami housing markets awesome austin awesome But around the country, there's some cracks in the foundation. This is allie wolf Uh, she is a chief economist at zonda She's involved in monthly meetings with the white house to discuss real estate trends and she says hey just so you know Our data is now capturing a 70 cancellation rate in phoenix when she's talking about his new homes So people are saying yeah, I'd like to buy that and they'll put money down and they're getting out of it because the rates are So high which makes a lot of sense, right? and then uh, some other people are You know chiming in entire Low estate market is collapsing much faster than anyone can imagine consider demand rate of plus 140 in 2021 now It's negative 27 percent New data will update in a few hours take a guess which way the numbers will be and that's okay I think we knew that was coming. That's just how it is and then also I thought this was good news, but maybe it's not good news of course the unemployment numbers came out and Unfortunately the fed wants unemployment to go up. That's just what they want because when unemployment goes up It's kind of hard because you won't have as much money that will decrease the man Of course inflation will go down. That's the theory, right? Let's see if that works out and the unemployment rates came out And the unemployment rate raises the 3.7 percent. I never thought I'd be happy to see that and I'm like, well, that's good news. Yeah I mean, that's what the fed wants. Well, kind of but not really So the fed also looks at a couple different things. First of all non-farm payrolls The expected was 195,000. They got 261,000. There's been an increase in payroll. Okay Unemployment expected 3.6 and went to 3.7. That's what they wanted, right? Average hourly wages year-over-years up 4.7 percent and also this little thing right here u.s adds 261,000 jobs in october unemployment rate at 3.7 percent so This would lead to this fed is seen sticking to rate rise After jobs report and I was like what? Employment rates up. That's awesome. No It is but it isn't the federal reserve has seen taking its benchmark policy rate upward after a government report Friday showed us employers hired more workers than expected last month. Jesus Louise. We can't win It also signaled some signs of slack and labor market And of course unemployment rate is what it is. So they hired more people For the unemployment rate went up Go figure So that's the bummer news, right? But just remember this Uh, this is ben's website. I steal from it as much as I possibly can call it into the cryptoburst you can Find a link in the description. It's valuable data. I can't show you all the good stuff But i'm basing essentially a large portion of when I sell out Or take profits on ben's website But there's a great one that he's got here. It's called the unemployment level And we talked about this a couple days ago and what you'll notice is that as the unemployment rate goes up See these little gray lines That's uh recessions And as you notice every single recession is followed by unemployment rates going up Well, isn't that bad rob? No, I'm gonna tell you why Because as unemployment rates go up.  I've been unemployed. I've been out of a job I know what it feels like. It's awful But for the broader thing scheme of things when we start to see a high unemployment and it peaks out Guess what happens? Recession goes away The fat gets trimmed the zombie companies get axed And the people that actually are there and and then the the stronger companies sort of say hey, guess what we're doing good things We need more people unemployment rate goes down And then of course it's a cycle again just like bear and bull markets happened here Happened in the 90s Happened again in 2000 the dot com bubble went up and it happened Bigly in 2007 same thing happened a little bit different circumstances And of course we can see that unemployment went to wow 15 15 14 000 filers are crazy And I'll begin here. We are so That's thousands of persons. I guess the percentage will be a little bit different, but you get the gist of it And to really put this into perspective remember Recessions don't last forever about a year and a half or so happened in the 70s 74 79 81 1990 2001 2007 But you'll notice that around here you had 10 12 years of economic Expansion here. You've got what six years Here you've got 91. I mean you got eight nine years or so So you usually have more economic expansion than you do have a retraction as far as recession What does that mean for me and you? I don't know if this is going to work out But it's looking pretty good these four year cycles If we're not in a recession already, some people say it is or not We're definitely going one in 2023. I think we can all kind of agree there But that's happens in 2023 and 2024 recession. When do we get out of it? 2025 so What happened in the four year cycles for bitcoin while there's a halving in 2012 you get an all-time 913 You get a big dip Next year and a reset in 15 same thing happened in 2016 You get a halving and you get an all-time higher next year a dip and a reset Same thing happened in 2020 you had a halving in 2020. We had this monstrous to all-time high I'm just kidding. It wasn't that big 67,000 we're going to the dip then we'll hit a reset and during this time in 2022 or assuming 2023 and 2024 We'll probably just be flat and have a recession what happens in 2025 and 26 we rebound and maybe I'll take that Now does that mean that's exactly how it is going to happen? No, but it's odd and it's eerie how things happen and Every single time I think I know where the market's going to go It just kicks you right in the face and says That's not what I'm going to do and that's why a dollar cost average because I can't time everything perfectly And right now the more I dollar cost average The more I like the whole approach. I'm doing that now since 2018 I know when I talk about taking profits and people like where's the profits to take? I maxed out on 2021. I you know went all in I gotta tell you the reason why I take profits along the way is because I can't I'm able to when I'm buying bitcoin at 18 Thousand 19,000 the days that it hits 20 21,000 I'm able to take profits the same reason that goes for ethereum when I'm when I'm buying it a thousand 1200 And it goes up to 1700 the same thing with polygon when I'm buying it's below a dollar 48 cents 43 cents I can take profits and then you might notice That the background is a little bit different today and the reason for that is because Uh, the house that we have in guainabo, Puerto Rico. There's no electricity bummer, but this was what it is Well, thankfully the condo in condado We stayed over there. That's our rental property and we were able to stay over there and that had electricity Until this morning when it just crapped out. So no electricity Now we're in our other condo Crossed away So again, not to brag Not to say anything, but diversity really helps out and it helps to take profits My goals are not your goals. It's just what you want to do and that's it. So look That's it for today. If you like today's news, give it a thumbs up consider subscribing I don't need to talk about our time sensitive If you want to stick around we're going to do the q and a right now I'll answer all your burning questions the best of my abilities, but trust me My knowledge is limited And we'll go from there and also remind you that the dca show will happen in 20 minutes right after this So you can go there and wait. I'll try to find the link for you And I'll put in the in the chat and just come over there and let the real brainiacs james and ben Talk to you about the macro events and what they think's going to go on. So that's it. All right. Let's jump into a little q and a