Welcome to toffee TV. It's the start of 11 show Tottenham Hotspere versus Everton Saturday three o'clock at the Tottenham Hotspere stadium very very tough game obviously Everton are Struggling a little bit with injury. So Yeah, I mean it's not listen It's still it's still gonna be a lot of the players that have been playing But there's gonna be a little bit of tiredness in there as well and not a lot of options coming off the bench It is what it is. Everyone's dealing with the same things at this time of year. It's always your struggle And you just have to get through it. Let's go with Jordan Pickford in goal Of course, he starts in goal at right back Nathan Patterson keeps his shirt There are no other options added by the sounds of it. So he'll continue at right back Yeah, and it's up just up to him to keep growing and keep getting better He has a few, you know, he does have moments where he switches off sometimes his positional play is not great But then sometimes his offensive play is really good and he gets he overlaps and he gets some good balls in the box So it's up to him to keep developing and keep getting better at left back. It's the retain of Michael Lenko The manager has said, you know, he's in contention and and that's enough for me to put him back in at left back And hopefully he gets back to the standard he was when he went out a couple of games ago the center backs James Tarkowski and Jared Branthwaite. Obviously Jared Branthwaite returned to the team on Tuesday night at left back But now he's backing at his His own position left of the center back. So That's the back four More my more traditional back four for us then we've had recently certainly three in there I've been playing most of the season and Nathan Patterson's played enough games. So that's my back four into midfield Going for two holder midfielders of Onane and Idrissa Gane obviously Ever now I'm gonna have a lot of the ball in this game talking about better of one the last couple of games And they are good a good attack inside the attack with a lot of pace and we're gonna need The fence of players in the middle of the park who can stop them and then take in the game round and get us going and I think with Disaganna gay obviously we know what he does. He presses really well And we don't on it in there. He's someone who sits in front of the back for very comfortable in possession Great tackle a good in the air. I mean obviously scored last week at Bailey Obviously listen, we all know what happened on Tuesday night. Hopefully he puts that behind them hopefully the fans put it behind them as well and Those two have a good productive game in front of them James Garner, maybe, you know, not really playing as such in the Decore a role, but Playing just in front of them, but still, you know still doing the work doing the doggies getting on the ball Which is his most important part of his game pressing from the front as well Well, I think I think that's why he got the nod Against Fulham because he does press so well from the front So I think obviously we lack a lot because the court isn't on the team But I just think for a game like this He's the one to step in we could have gone with two strikers We could have maybe brought Dan Tumminen, but I just think for this game I can I can see having more like a three in midfields it and stop sort them playing and then You know spring them with the wide men. So that's my midfield sent in the central area on the right-hand side Jack Harrison And again last weekend's Baylor thought was great as the defensive side of his game But also the offset offensive wasn't as good against Baylor So against Fulham on Tuesday night Um looked a bit leggy, but again not lots of options So I'd start with him on the right on the other side Dwight McNeill will continue again Doty did excellent in Baylor, but again Fulham not so good Still work the socks off, but just not that creativity Especially when he swapped to the other wing So we're going to need loads out of those two in this game with having three in midfield who are um More combative, let's say And up front i'm actually going to go with Beto for this game. I thought Dom looked a little bit off it on Tuesday night and Beto's got He's been scoring goals recently. He's got you know And that's what you can judge them on at the moment. I think maybe just that little change might give us a little spark And it's not always about going long With a ball over the top Sometimes you need to go down the channels and I think with Beto you've got a wheel and runner that can get us up the pitch What we don't want is in the game is the ball just coming back all the time and being under pressure I know listen. We've been fine out of possession But I think sometimes when you're tired It becomes a mental thing as well when you keep on just turning it around all the time A dom just looks a little bit offy dom being on the bench would be great You know, I don't think he's been on the bench since the brimford game I think in that game he came off the bench and scored and that's so Him the ability for him to come off the bench would be huge But I think Beto I think he just does it. I think he deserves a start I think this would be a good opportunity for him Um, and then you've got dom too can come on and maybe if dom can come on Maybe those two then play up front together for a little spell in the game Because you don't maybe aren't looking at the same way Whereas when dom starts and the manager looks at him thinks gotta be careful with him so That's my 11, of course Obviously I haven't playing on saturday and tuesday and we play again next wednesday and next saturday Gotta protect the players. So that's the 11. I'm going for let me know your thoughts in the comments What would be your start 11 for this game? Would you be looking at dan duma? Would you look at dobbing? Would you play a three at the back? Um Which could be a good way to go as well in this game Let me know your thoughts in the comments. Make sure to give this video like Make sure you check out the match preview Subscribe if you haven't already if you want more great videos join us over on toffee tv premier The link is in the description and the qr codes come on the screen now. See you later


OK, so let me begin by just reviewing where we are in the whole construction. So we start with some moduli space, or system of moduli space of pseudo holomorphic curves. And then on this moduli space, or collection of moduli spaces, we construct some extra data, which I'll call an implicit atlas. And then what we'd like to do is, from that implicit atlas, construct the virtual fundamental cycle on the moduli space, or on the virtual fundamental cycles on the system of moduli spaces. Now, one important aspect of these two arrows is that both are canonical. So recall last time I described a construction of implicit atlases on the moduli spaces relevant for contact tomology. But if you understand what's going on in that construction, you see that it doesn't actually use anything special about being in the situation of contact tomology. It really just applies to any sort of reasonable moduli space of holomorphic curves. And moreover, what it produces is an implicit atlas, which didn't depend on making any extra choices, as it's canonical. Now, OK, canonical, I don't mean unique. I mean, of course, you could find some other way of defining an implicit atlas. I just mean that in the process of making this definition, I didn't have to make any arbitrary choices. And similarly, there are sort of many ways you could try to go from an implicit atlas to a virtual fundamental cycle. But I'm going to describe a way which doesn't involve making any extra choices. It's canonical. So today, what I want to do is I'll give you a complete definition of an implicit atlas. And well, as you probably know, there are somehow lots of other things one can try to put here in particular Korean-ish structures, others as well, which I mentioned last time. And so I'll try to say what some, I'll point out some features of this definition of implicit atlas, which I think make it this very close to the midpoint of the shortest path between the left and the right. OK, so I'll define implicit atlas. I'll point out some of the good features of the definition, which enable these canonical constructions. And I'll tell you how to construct a virtual fundamental cycle out of an implicit atlas. Moreover, it'll be a canonical construction. So I mean, I'll literally just write down in terms of the charts of the implicit atlas a chain map, which represents the virtual fundamental cycle. So first, let me just remind you of the basic idea. We have some modularized space, x, which is up priori some horrible set, just compact house dwarf. We find dimensional reduction consists of the following. So it consists of an open subset, u alpha inside x. And it consists of a chart, which I'll call x alpha and refer to as the alpha thickened virtualized space. And an embedding of u alpha into x alpha as the zero set of some function to a vector space, a finite dimensional vector space.  Now, the reason we want to equip x with charts like this is because they tell us what the virtual fundamental cycle should be.  So how lying close to x and that manifold, at least locally over u alpha, represents the virtual fundamental cycle of x. And now the key problem where we have to solve in the second arrow is to figure out how to glue together these local virtual fundamental cycles into a global virtual fundamental cycle over all of x, so just given this local data and certain local compatibilities between the data. So let me now define for you what implicit at this is precisely. x will always be compact Hausdorff. And a is just some set, usually an infinite set. It will index these charts. So I'm going to push to that list.  So let me just list these. So first, there are what I'll call covering groups, gamma alpha for alpha in A. These are finite groups. And just some notation for finite subset of A, rather, gamma I denote this product, just notation. In covering groups, then we have destruction spaces, E alpha for alpha in A. These are finite dimensional vector spaces. And again, EI will be the direct sum of E alpha for alpha in A. So more over, gamma alpha acts linearly on E alpha. So E alpha is a finite dimensional representation of gamma alpha. Thirdly, there are these thickened spaces, x of i. These are just Hausdorff topological spaces for finite subsets of A. And the original modular space x is identified with the thickening corresponding to the empty set.  And the only way it somehow interacts with the space x we're interested in is through this identification. So first, there are the Karinishi maps, S alpha from xi to E alpha for alpha in i continuous maps. Second, there are footprints i, j contained in xi open subsets for i contained in j contained in A. Six, there are these footprint maps, cij from zero set of sj minus i on xj to uij for gamma j equivariant. Of course, gamma i acts on this as a gamma i invariant open subset. And we just save gamma alpha for alpha in j minus i extravially here. So this is the footprint maps.  So xi is some modular space. It's contrary to what I've drawn here, it's not necessarily everywhere cut out transversally. So we have to mark gamma i invariant subset where it's cut out transversally. So there are a bunch of requirements for this data, but let me just pause and somehow relate it back to this picture to make sure we're all on the same page. So we have x. So suppose we have, let's do an example, and close that with just a single chart. What does that mean? So we have space x. That is the same as x empty set. There's a space x alpha. It's a thick of modular space. And there's a map from s alpha from x alpha to e alpha. So I'll write x alpha, but maybe I should really sort of put brackets around alpha. I won't. Inside x alpha, there's this zero set, s alpha inverse of zero. And this footprint map identifies, or gives a map, from s alpha inverse of zero, it'll be an axiom which says it's a nice morphism, onto some open set, which is u empty set alpha inside here. So this is exactly what we have over here. I realize now I forgot to explain some notation. What is this? I only defined s alpha. So as before, si will be the direct sum of s alpha for alpha in pi. Now there's a little bit of ambiguity in this notation because there are many functions called s alpha because there are various possibilities for i here. And so that's why I have to specify the domain here. I take the direct sum of all the s alphas for alpha in j and not in i, and look at the action of that on x j. So this is the picture I get on one alpha. It's just a single element, which is more or less what I wrote over here. The difference is only here I required x alpha to be a manifold. Here that requirement, we haven't seen that requirement yet, but it'll come later. And finally, just the other thing to say is that in this picture, I assumed gamma alpha is trivial, but of course, gamma alpha is non-trivial. There's a similar picture. We have actions here. This map is aquavariant. And now this is not an isomorphism, but just a quotient map by gamma alpha. This is a gamma alpha cover. So what are the axioms for this data? So there are lots of these axioms, but most of them, as you'll see, look rather tautological, and they are. So in any particular, so first of all, there's just some universal construction of an implicit atlas on any reasonable modulized space of holomorphic curves. And when you do that construction, almost all of these axioms are trivial consequences of the definition. There's only a few that you really have to carefully. OK, so up here, 6 was just some map, but now let me sort of be more precise. We actually demand that this is a homeomorphism after quotienting the domain. So this is one of the axioms, which is somehow requires a little bit of argument in practice. Not so serious, but there's something there. j x i rag is contained in x j rag 7 gamma j minus i acts freely on here. Now finally, some axioms, which are very important. So this should be open. As the locus where this modulized space is transfer should be an open subset. Second, s j minus i from x j to e j minus i should be locally modeled on this projection over the inverse image of the regular locus. So in particular ways this mean, if I take j equals i, then the target is just a 0 vector space.  So let me just write this. x implies, as a special case of axiom 9, x i rag is a manifold of dimension d plus dimension e i. And moreover, whenever I have some, say, x i and x j, and I have some point in x i, which I declare to be transverse, when you view that point as being cut out inside x j, the corresponding section by this section, this section is transverse over that locus, locally modeled on this. So here I'm only demanding topological condition. This is locally modeled topologically here. This is a topological manifold of dimension, d plus dimension e i. Yeah, of course, there's an analogous definition, analogous notion of an implicit Alice with a smooth structure. You could require things to be smooth. The point here, though, is that for this second arrow, we don't need a smooth structure. I'm going to give a definition of the VFC via some algebraic topology, and it won't actually use any smooth structure on these thick and transverse modulite spaces. And this first arrow is easier if we don't have to construct a smooth structure on these spaces. So these axioms require some argument in practice. Finally, last axiom, which is also not important, is that this space should be the union of the transverse loci. So if I look at the transverse x i rag, this gives me, this gives me, or via, is some subset of x empty set via this footprint map. And these subsets better cover all of x empty set. This is basically just saying, somehow, we have enough charts to cover all of x with transverse thickenings. Obviously, there's no hope of recovering the virtual fundamental cycle if these thick and modulite spaces, the locus of a transverse dough doesn't cover all of x. So that's what we're requiring here in axiom time. So this looks like a lot of axioms, but I should just remark that in practice, we have this canonical procedure for constructing one of these. And in practice, there really only two things which require some sort of argument. The first is this. You have to show that this procedure gives you enough charts. In the case of contact tomology, and usually in general, what this amounts to is just saying that any curve in your modulite space that you carve out, you can stabilize it by intersecting with a transverse divisor. That's somehow the ingredient which goes into this. Five is actually, maybe I should erase this. Five is not actually all that hard. It's something about comparing Gromov's topologies. For eight and nine, this is some amounts to a gluing theorem. So if you want to prove eight and nine near a point where the domain of your holomorphic curve has no nodes, then it's easy. This is something that just follows in the implicit function theorem for Banach-Menthefels. Neighborhood of a nodal curve, you have to work harder. This is where the analysis comes in, verifying eight and nine. So just briefly, the example you should keep in mind is x is just some modulite space of maps u, del bar u equals 0. Then thickening xi is some space of maps u, along with choices of e alpha in e alpha for alpha and i, such that, so first, u satisfies some open condition depending on alpha for all alpha in i, and del bar u plus e alpha equals 0. So this is just sort of a cartoon version of the definition I gave in the previous lecture. But even given just this cartoon version, you can check that most of these axioms, things like this, are just sort of tautological. xi reg here is a locus where xi is cut out transversally and has trivial isotropy. That is the automorphism group of the particular curve with whatever marking we've chosen in addition here is trivial. OK, so some remarks about this definition. The set A is, in practice, the set of all thickening, I call the elements of a thickening datums, is some universal set of all possible choices you might make of a thickening datum. So it's a very infinite set, uncountably infinite set. And one thing to point out is it's somehow, it's not completely a triviality to formulate a definition which allows you to make thickened modulized spaces for arbitrary subsets of thickening datums. Remember, if A is a set of all thickening datums, somehow I could, I'm demanding here that I can construct these thickened modulized spaces even when I pick two thickening datums which are very badly non-transverse to each other intersecting simultaneously with two divisors which are not transverse to each other. And the fact that we can formulate a definition which allows this sort of universal construction taking into account everything at once is not entirely trivial. And it's somehow not, it's not obvious in other constructions that this is possible with other notions of atlases. Another sort of difference between this definition and others is that XI, OK, so one is that XI reg, you might want to require that this XI is transverse everywhere for non-trivial thickenings. So as long as we thicken at least some, then we get a transverse modulized space. So this is tempting to include this as an axiom because of this picture right here. This X alpha I drew as sort of transverse everywhere. And this is the case which is most intuitive. On the other hand, this requirement makes it much harder to construct an implicit atlas. And it doesn't let us do lots of other things we want to do. So I'll give an example. So let's just say we don't want to include this axiom. So more generally, these XIs are treated on equal footing independently of the cardinality of I. So in other definitions of similar to this one, you have one sort of formalism for how X empty set is cut out inside X alpha. And you have another formalism for how XI is cut out inside XJ when XI is non-empty. And this somehow leads to sort of duplication of the axioms that you have to check. So this is, I think, a good simplification. Another reason why both of these points are very important is that there's this construction of the product implicit atlas. So if I have space X with an implicit atlas A, space Y with an implicit atlas B, I can construct an implicit atlas on the product with index at A disjoint union B. Namely, the thickened moduli space X cross Y thickened by I and I prime is just the product of thickened moduli spaces. And the regular locus, well, there's sort of only one reasonable possibility. It's XI reg cross Y I prime reg. OK, now we see immediately that this construction, even if you started with implicit atlas as satisfying this property, the product doesn't necessarily satisfy this property. Because I disjoint union I prime could be non-empty, but one of I prime could be empty. It's also clear that this somehow completely fails if we don't treat XI as an equal footing for X empty and non-empty. So some people find this notation quite strange. They use the same notation for the moduli space and the thickened moduli spaces. But I think this is a good justification for why I think that's natural notation in this particular setting. Right. OK, so finally, this is the definition for somehow moduli spaces without boundary. I require the thickenings to be manifolds. Of course, there's a natural generalization of this implicit atlas with boundary. If you have a space X with closed subset known as its boundary, or just called its boundary, there's a notion of an implicit atlas on this space, which just consists of thickenings inside each thickening re-we specify closed subset. And then all of the axioms are the same, except the local model we now allow is a manifold with boundary as a local model. And in the local model, somehow the boundary of XI reg as a manifold with boundary should coincide with this subset that we declared to be its boundary. And you can do the same thing with more general stratifications. Just a matter of requiring that the stratification be compatible with the footprint maps and specifying the right local model that you want. So last time I constructed implicit atlases like this on the moduli spaces for contact technology. So now I've given you the definition. Last time I gave you the construction. Now I want to give you the construction of the virtual fundamental cycle. So as I said earlier, what I'm going to do is just simply write down a chain map which represents this virtual fundamental cycle on the chain level. And this construction will be canonical. In particular, it's functorial with respect to inclusions of boundary, inclusions of other strata in implicit atlases. And as a result, it'll work well for homological theories where we really need a theory of somehow coherent virtual fundamental cycles over a collection of many open sets, of many moduli spaces. So VFC machinery. So this will be a compact halter space. And A will be implicit atlases on X. I'm going to assume for convenience that A is finite right now. This is mostly for notational convenience. Everything I'm saying works for infinite A simply by taking a direct limit over finite subsets. So this isn't a big assumption. Plus, choosing a finite subset of a given atlas isn't very much of a choice. So the first thing we have to get out of the way is orientations. So if we have an atlas A on X, we get an orientation sheaf. OX on X. It depends, of course, crucially on A. I won't write that dependence here. It's stock at a given point P in X is given as follows. So we choose a lift, P tilde in XI reg. And then the stock at a particular point is the orientation sheaf of XI reg. Remember, that's a manifold. So it has a well-defined orientation sheaf. Stock at P tilde tensored with the orientation module of EI dual. Finally, we take invariance under the stabilizer of P tilde under the action. Again, I. So we say this atlas is locally orientable, even only if this invariance is always non-zero. So our orientation modules are always isomorphic to Z. The orientation module of a vector space is a free Z module of rank 1. And choosing a generator of that copy of Z is equivalent to choosing orientation on a vector space. So that's what orientation modules are. This action, crucially, is just by a sign. So if gamma I is trivial, or if I'll have odd order, you always have local orientability. But if not, we need to impose this condition in practice. It's always satisfied where the modular spaces we're interested in. So henceforth, we'll just have this as an assumption that X is locally orientable. So this is just a local system on X. And of course, the virtual fundamental cycle will be in homology twisted by the dual of the orientation sheaf. So what's the goal of this VFC machinery? Well, it will be to define the following structure. Actually, maybe I'll write it here. So the VFC machinery will consist of the following. So we will define, first, a chain complex, which I'll call virtual co-chains on X with respect to A. This is just notation. I'll say what this is. I mean, the main thing I'll say is, what is this complex? It'll come with a natural push forward map to chains on a tom space of E. It will be canonically Clausi-isomorphic to check co-chains on X. So this is what we'll construct. After making this definition on orientations, now I'm going to leave orientations out of the notation. Really, I should twist this by the dual of the orientation sheaf of E. And this should have local coefficients in the orientation sheaf. But for simplicity of notation, let me leave this off. So this is what we define. Why is this the virtual fundamental cycle? So this chain complex is very easy to understand. It's the tom space of EA. This is just EA is a finite dimensional vector space. This is a sphere. And so this is just concentrated in degree D. So Z concentrated in degree D. And what we have, or up to Clausi-isomorphism, is a map here. So i.e. what we have is, so you get a cycle, denoted to x, v, in the dual of check co-chains. So let me write it over here. Z in degree D. This is check co-chains. X, we have this map. What this is is simply an element in the dual of check co-chains in degree D. So as an aside, this is check co-chains. And recall what is this. This is a direct limit over open covers of co-chains, the nerve. Now, what is the chain level dual of this? Denote the chain level dual by C upper bar. And it's called Stienrod chains. This is used much less than check co-chains. And I'll spend most of the construction just in the check world. But it's somehow the final answer can be understood this way. Stienrod chains is not the dual of check co-chains, which is the same as a derived inverse limit over open covers of the homology of the nerve. So this is somehow the right way of doing homology in the check sense. I mean, you can also take the homology of the nerve and take just the usual inverse limit of the homology of the nerve. You get something called check homology, but it's not a good object to consider. This is better. So for probably wondering why, we have to use all these exotic homology and co-homology theories. Well, so the exotic is not meant in the technical sense. I just mean all these strange co-homology theories. So the reason is the following. So the virtual fundamental cycle, unfortunately, does not lie in general. It does not lie in singular homology for the following very simple example. So let's take this space, w, as the Warsaw circle. Let's cut out inside r2 by a function, which is 0 on the inside, negative on the inside, positive on the outside, s from r2 to r. Now, clearly, the virtual fundamental cycle of the space should be non-trivial. It goes around the annulus, goes around the loop. If you push it forward to a regular neighborhood, which is homomorphic to an annulus, then you get something non-zero. Unfortunately, w singular homology is 0. Virtual fundamental cycle will never lie in usual homology. But Stienrad H1 is indeed z, as you think, generated by going around the loop. So now our goal now is to define this object and define this diagram that will give for us this virtual fundamental cycle on the chain level. So let me first just say, how do we do this for a single chart? That's already not obvious. And then we'll generalize from a single chart to multiple charts, a single chart. So we want to consider the following diagram. So let's look at chains. So recall that the dimension of x alpha reg is d plus the dimension of e alpha, where d is the virtual dimension. So now we have chains x alpha reg, x alpha reg minus x. So just consider the chain group. So let's first let's recall that x alpha reg, that's a manifold, we have a map to e alpha, and x is simply the zero set of this map. This is a global viantimensional reduction. So I just want to say how this works in the global case first. We'll assume that x alpha is cut out transversely everywhere. So now there's a push forward map, s alpha push forward, going to chains on e alpha, delta to zero, the time space of e alpha, which, as we said before, is quasi-isomorphic to z in degree d. And now by punctual duality, there's a quasi-isomorphism between this and check co-chains on x. So what we get is a map like this. And I claim this is the virtual fundamental cycle. So let's just discuss the smooth case first, where s is the smooth section, maybe even transverse to zero. This is not entirely obvious, even in that case. It is true. It's the assertion that this is the virtual fundamental cycle is very similar, in fact, basically equivalent to the assertion that the two standard ways of defining the Euler class of a vector bundle are the same. Namely, you can either take a generic section, take the fundamental class of its zero set, or rather the Poincare dual of the fundamental class of the zero set of a generic section, or you can pull back the Tom class from the fibers. This definition of the VFC is similar to the pulling back the Tom class. Basically, it is pulling back the Tom class. And this is a relative Euler class. And the usual perturbation picture where you perturb s alpha, take the fundamental class of the zero set is analogous to the corresponding definition of the Euler class. So in this case, oh, we get the definition of the VFC and the degrees with the definition that you're used to. So in this case, this complex right here, this is this virtual co-chains complex. So this diagram I wrote down is exactly what we wanted. So now let me say how to generalize this to multiple charts. Again, let me just do this very explicitly in a single case of two charts and an overlap chart. Although what we do for this will somehow, say, exhibit all of the complexities necessary to deal with the general case. So first, let me try to draw one of these things. So here's our space x. And I have one chart, say the alpha chart, which looks like this. So we have a manifold. So again, I'm going to assume that all the non-trivial thickenings are just regular everywhere. So manifold x alpha, which is the same as x alpha reg, maybe I'll just write x alpha reg. And there's a function from x alpha reg s alpha to e alpha, the zero set is identified with some open subset. So that's one chart. It's the same as what happened over here, except it just takes place over some open piece of x. Now I have a similar thing happening on another piece of x. So this is x beta reg. And then we have exactly the same situation. There's a map, s beta to e beta. And s beta inverse of 0 is u beta inside x. And x is covered by u alpha and u beta. Now, our atlas consists of another chart, namely the alpha beta chart. Indeed, as it should, because somehow the alpha beta chart is exactly what we need to glue together the local virtual fundamental cycle from the alpha chart to the local virtual fundamental cycle from the beta chart. So what does the x alpha beta chart look like? Well, it's some higher dimensional manifold looking something like this, x alpha beta. It has two functions, s alpha to e alpha and s beta to e beta. If I look at just the zero set of s alpha, that gives me an open subset of the x beta chart. I look at the zero set of s beta, gives me an open subset of the x alpha chart. These are both cut out transversally. So on x alpha beta, I have two sections. Both of them are transversed to 0. S alpha cuts out x beta, or rather open subset thereof. S beta cuts out x alpha, an open subset thereof. Now if I look at s alpha direct sum, s beta, this is now an open subset of x, namely the footprint of this alpha beta chart is exactly the intersection of the footprints of the alpha chart and the beta chart. And that's not necessarily transverse. So s alpha and s beta cut out transverse submanifolds, but these submanifolds don't necessarily intersect transversally. Their intersection is exactly x. Or, again, rather of an open subset of x. So this is the picture. So now let me say how do we want to glue together the virtual fundamental cycles. So the first try is the following. So let's take, again, chains d plus convention e alpha on x alpha reg as before, direct sum chains on x beta reg. So why are we doing this? Secretly, this is Clausiasomorphic 2. This gives me check co-chains with compact support on the footprint of the first chart. And this gives me check co-chains with compact support on the footprint of the beta chart. Now we have to glue these together using information from the alpha beta chart to do this. Oh, we take a mapping cone. So we look at chains d plus convention e alpha beta x alpha beta reg x alpha beta reg minus x. So we take some map. So what I've drawn here is a map of chain complexes. What I mean is you take the cone. So now this is Clausiasomorphic 2. It's Clausiasomorphic 2. Check compact squared, check co-chains on the intersection. This mapping cone, which is Clausiasomorphic 2, check co-chains on x. Again, compactly supported, but OK, x is compact. So it's the same. So this looks very good as a replacement for this complex. Unfortunately, defining this map on the chain level is rather cumbersome. So sort of intuitively, what's this map? It should be the cat product with the savings map as cat product with S beta pullback at Tom class. But somehow making this on the chain level is possible, but it leads to a mess. So we don't want to do this. We want to do something a little bit different. So the problem with this construction basically is that the dimensions differ between these charts. So to fix that, let's just take the alpha chart and cross it with, there's not enough room here. Let's do it somewhere else. So to fix this, we'll just take the alpha chart, cross it with e beta, and we'll take the beta chart and cross it with the alpha. Now everything's the same dimension, and things should go much better. So we're continuing this example. So virtual co-chains will be defined as follows. Let's start sufficiently far to the right. Chains d plus dimension e alpha beta on. So we take x alpha cross the beta chart, and then we take the complement of x cross 0. We write complement like this. And we also want chains e alpha cross x beta relative to the complement of 0 cross. I'm leaving room for something in the middle of which we'll use to glue them together. So how do we glue them together? We're going to do a deformation to the normal cone. So let y alpha beta be the following space. We look at x in x alpha beta, e alpha in e alpha e beta in e beta, and t in 0, 1, such that s alpha x is t e alpha, s beta of x is 1 minus t e beta. So this space has a natural map to 0, 1. I think there's a family of spaces over 0, 1. So to understand what it is, let me just tell you what the fibers are. So if t is strictly between 0 and 1, then you can divide by t. And you can divide by 1 minus t. Include that e alpha and e beta are determined uniquely by x. So for t between 0 and 1, the fiber is just x alpha beta. For t equals 0, you can still divide here. But now this equation says that s alpha must be 0, and e alpha can be anything. So I get e alpha cross the 0 set of s alpha, which is the beta chart, or rather an open subset of the beta chart. Similarly for t equals 1, I get u alpha alpha beta cross e beta. So this is called a deformation to the normal cone. And so why is that? Well, so you think of it the following way. As t goes to 0, we're blowing up along the sub manifold u beta alpha beta. So you have a manifold x alpha beta and a sub manifold, given by the 0 set of s alpha. And as t goes to 0, we blow up along in the normal direction so that when we actually get to t equals 0, all that's left is the total space for the normal bundle. So the motivation for this construction is that we'd somehow like to pick a tubular neighborhood, but there's no canonical choice. However, there is a canonical deformation. There's a canonical deformation from x alpha beta to the total space of the normal bundle. And this didn't require us to choose a tubular neighborhood. And so instead of choosing a map from the tubular neighborhood, we defined a family containing both. So how do we use this space? So now we'll get y alpha beta relative to the complement of x cross 0 cross 0 cross 0 1 complement. And now we'll glue together again using a mapping code instruction. So I use, say, u alpha alpha beta cross u beta complement of x cross 0. So this maps in here just by inclusion. It maps to here by the t equals 1. And similarly, here with the alpha and beta swapped. So this whole complex now is, by definition, virtual co-chains on x with respect to its Alice. And as before, you can write down this isomorphism. This is check co-chains on u alpha. This is check co-chains on u beta. This is check co-chains on u alpha intersect u beta, all with complex support. And you get this. This is quasi-isomorphic to check co-chains. So I'll say a little bit more about how to make this quasi-isomorphism precise very soon. So I defined now the virtual co-chain complex. There's a canonical push-forward map from this to chains on the tom space of e alpha beta. Namely, we just take the obvious push-forward map on all these spaces. And then the degrees and the inclusions of these. So you just take that push-forward map here and 0 here. That gives us this map. So this is the definition for two charts. Just for illustration purposes, let me write down the definition in general. So for general x and a, you define virtual co-chains on x with respect to a is the following. So direct sum over p, direct sum over chains of finite subsets of a of chains of dimension minus p plus d plus dimension set e in ea t in. So this is some complicated definition, which is a natural generalization of what I wrote up before. Just the only thing you have to do is figure out how to organize everything when you have many charts relative to the complement of locus where. So here's the definition. So this is some rather complicated chain complex. What you gain here, though, is that this definition of the virtual fundamental cycle, one is functorial. It didn't require any choices. And second, it works without having a smooth structure on a thick and marginalized basis. And since we are just using algebraic topology to define things. The other remark is that this is somehow sort of homotopy co-limit of chain complexes. So what's the deal with homotopy co-limits? Well, so a lot of times when you want to try to define the virtual fundamental cycle, you take some big, this joint union of the thick and marginalized basis, maybe the regular loci, and divide by some equivalence relation. Now this is a very, sort of modding out big equivalence relation here. It's a very drastic thing to do, because there are simple questions about this topological space, very basic properties which require a certain amount of work to verify and to ensure. I mean, in some cases, they just aren't true if you do this for certain atlases that might fail to be housed or formatizable. So this is an example of a co-limit. Just glue things together naively. If you instead do sort of homotopy gluing by taking a homotopy co-limit, sort of automatically has good properties. And you don't have to worry about this. And this is sort of an example of a homotopy co-limit. So you could take this construction and just forget about chains. Just do this at the level of spaces. Glue together these spaces according to this simplicial diagram. And you just view this as a simplicial space. And this automatically has good properties. It has the properties you want to use it to define the universal fundamental cycle. You don't have to mess around with the atlas at all. OK, so just aside, let me just give a basic example of this difference. If you have some, say, two spaces, x and y, and you have a common subspace. Between them, you can take the dysteroenia of x and y, modulate this equivalence relation which identifies them. They say open subspace.  This is the naive columnit, homotopic columnit, which is better to take x union, y union, u cross in interval and say that if u in x is equivalent to u cross 0 and u in y is equivalent to u cross 1. So that is I take the u cross 0, glue that to x, take u cross 1, and glue that to y. And now this is a much better space to work with. And the point is that this is what you really wanted anyway. So I defined this virtual fundamental cycle on the chain level, just some canonical construction. One thing which I didn't explain, which I want to explain now, is how do we show the isomorphism between this, the quasi-isomorphism between this complex and check co-chains on x? This is, I gave a sketch of why it should be true morally using point of duality. But now let me say in more detail how this actually goes.  For when you're talking about sheaves on compact spaces or locally compact householder spaces, it's often convenient to also have an alternative definition of a sheaf which uses open sets instead, which uses closed sets instead. So we say a k sheaf is a functor f from compact subsets back to abelian groups, such that following axioms are satisfied, the f of empty set is 0, f of union is left exact, and 3, this is an isomorphism. So now you get this category of k sheaves on x, and it's equivalent to the usual category of sheaves on x.  And if you have a sheaf, to get sheaf on compact sets, you take the direct limit over open neighborhoods. So that's the correspondence. Now there's something called a homotopy sheaf, which I'll only define in the k setting. So a homotopy k sheaf is a functor f from compact subsets of x to chain complexes with the following properties. So first, f of empty set is acyclic, f of k1 union k2, f of k1, f of k2 to f of k1. Magnetite k2 is exact, meaning the total complex is acyclic, and this direct limit is a quasi-isomorphism. So this is a homotopy k sheaf should be thought of as a complex of sheaves, which calculates its own cohomology. So for example, any complex of soft sheaves is a homotopy sheaf. So a sheaf is called soft if and only if all of the restriction maps on compact sets are surjective. So another example of a homotopy sheaf is k maps to singular chains on x relative to the complement of k for any topological space x. This axiom is obvious. This axiom amounts to Myroviatoris, and this is some compactness of simplices. So this proposition, so I said in words, homotopy k sheaf is something which calculates its own cohomology. Let me write it precisely. Then there's this natural map from f to check cohomology of f, and this is an isomorphism. There's a natural map here for any complex of pre-sheaves here, and if the complex of pre-sheaves is a homotopy sheaf, then this is myomorphism. So why is this machinery useful? Let me give a non-trivial application here. So here's what we're really interested in. So as opposed to f, this homotopy k sheaf is pure, i.e., the homology of the stocks is concentrated in degree 0, and the cohomology on any compact set vanishes for i sufficiently negative. So one is somehow the key condition, two is technical and might not be even needed. I don't know. So then 1h0f is a sheaf, and 2, the homology of f, is naturally kinetically isomorphic to the check cohomology of that sheaf. So given this proposition, what this is really just saying is that if f is pure, then it's a quasi-asomorphic to a sheaf. So f should be thought of as an injected resolution of some underlying sheaf, h0. This all looks very abstract, but here's an example. It's really the main example we care about. So let m be a topological manifold of dimension n. Then you look at k maps to chains on m relative to the complementary k. And this is a homotopy k sheaf, as I remarked before. And now using the fact that m is a manifold, we can show that it's pure. This is just the standard calculation of the homology of r, in retrospect, the complement of a point. And now, thus, by this lemma, we conclude that the homology of this homotopy k sheaf is isomorphic, canonically, isomorphic to the check cohomology of k with coefficients in the associated sheaf, which is simply the orientation sheaf of m. So this is point creativity for arbitrary closed compact subsets of a topological manifold. This is clearly a non-trivial interesting state. So this is how this machinery is useful. Now, how is this relevant for this? Well, there's a natural extension of this virtual co-chains complex to be the global sections of a homotopy k sheaf on x. Namely, you simply require here, basically, to do the analog here instead of taking the complement. This is sort of the complement of all x. If you want to say what are sections over a closed subset k, you take the complement of where the x lies in k. And now, this is a homotopy k sheaf. And you can check that it's pure. And this gives you the identification you want of this with check co-chains on x. So this completes the construction of the VFC machinery in the case x has no boundary. And as you saw, we just wrote down some big complex and the map of complexes. And that was the virtual fundamental cycle. So now, at the end, quickly, let me just say how this works in the case with boundary. Because somehow, that's what we're really interested in. Case of floor homology is much more interesting. So first, so with boundary. So let's say x has an implicit Alice with boundary. I already said what that is earlier. In particular, you get an implicit Alice on without boundary on the boundary of x. So now, our earlier construction, a construction definition I just erased, it does not give something which deserves to be called virtual co-chains on x. It gives something which deserves to be called virtual co-chains on x, rel the boundary. So this is somehow, the reason we get rel boundary is for the same reason that in the unusual Planck-Riduali what happens, you have the homology of the boundary of a manifold going to homology of the manifold going to homology of m, rel the boundary. This is isomorphic by Planck-Riduali to co-homology of m. This is isomorphic to co-homology of m, rel the boundary. And then there's this restriction map, hn minus star of the boundary. And this corresponds to the connecting homomorphism here. Connecting homomorphism corresponds to this just restriction on co-homology. And the push forward map here corresponds to the co-boundary map on co-homology. Just somehow, trading boundaries is what happens. This is why we get something, rel boundary. So now there's a map, right? So there's a map from virtual co-chains of the boundary to virtual co-chains of x, rel the boundary. And the fact that it increases degree by 1. We have isomorphisms between this and check co-chains on boundary x via the co-boundary map to check co-chains on x, rel the boundary. So this is just some natural push forward map on singular chains. And what it gives us under these isomorphisms is a co-boundary map. Now we have maps from this to z in degree d. This diagram commutes just simply, basically, by construction. This diagram commutes by construction. This is some non-trivial statement which you prove using the Schief theoretic machinery, which is used to define these vertical isomorphisms. So what does this tell you? Well, this map here is the virtual fundamental cycle of x. This map here is the virtual fundamental cycle of the boundary of x. So what this tells you is that we have this boundary map on chains. And this sends the virtual fundamental cycle of x to the virtual fundamental cycle of the boundary. So this is important, for example, if you want to, even in the most basic settings where, say, you want to prove Gromov-Witton invariance are independent of the almost complex structure, you take a path of almost complex structure, and you have a modular space with boundary. One boundary being the modular space for complex structure number 0. And the other boundary being the modular space stable mass for complex number 1. Complex structure number 1. And this fact that the total space can be given the virtual fundamental cycle, whose boundary is the difference of the two sides, it gives you this independence of j. A bit more sophisticated is that you can, this construction somehow, we did everything at the chain level. So this diagram allows us to have a notion of a coherent collection of virtual fundamental cycles over a bunch of different moduli spaces, which is what we need to do for our homology. And this also extends for the case with manifolds with corners. Just you have to draw bigger diagrams, that's all. So finally, let me just say there's also a product map. You have two spaces you can look at. So we really wanted this nice definition of the product of two implicit atlases so we can make a construction like this. This is isomorphic to check-co-chains on x. The whole boundary, and there's cunith-causiasomorphism here to check-co-chains on x cross y. Real boundary, and this commutes, again, by some sheaf theoretic considerations, which are used to define these maps. So in this compatibility of virtual fundamental cycles with products, probably what this gives you is, x times y, for is x for y, for this compatibility on the chain level with products is also crucial for floor homology. Because you want to be able to say that the virtual fundamental cycle of the boundary of a moduli space is expressed as a product of other moduli spaces. You want to say the virtual fundamental cycle of that boundary is the product of those virtual fundamental cycles on the chain level. OK, so this completes the really definition of the VFC machinery. Next time I will show how to apply this to construct the virtual fundamental cycles for contact homology given the atlases that I constructed last time. I'll also review a little bit the definition of the atlases.


Okay, good morning, everyone. How's everyone doing today? So basically, I have a thing I have to pull on. I think we should do a new video. I'm as crazy as it sounds to go long when we do have some shorts.  I did look at Babba, it doesn't rate so good, but if the market falls today, Babba will go. Babba will go. The crazy thing is this market, again, it's bullish, okay, it's bullish. We're in an uptrend. We never broke the uptrend really, not even last year, but I think we've just expanded the range, we'll see. So I'm in options that are calls, inputs this week and next week. I mean, I'm in a mix of things that I just like, either up or down. So it's kind of a bizarre time right now to be trading, but you really have to be looking for specific, specific, specific things to do. I mean, the video is just higher. So it's, again, this is not earnings or anything like that. It was lower this morning, now it's getting bought.  So based on the price point, if you wanna buy a call today, you could just, if we do this and you don't wanna do it as an equity trade, just buy the 430 calls, I will look at, I will look at the target. If for some reason this doesn't set up, I will look at BABA if this doesn't set up. But, you know, this is very strong and we will talk about the market when we're done. If my clock is off, a knock on wood. I've had no internet problems. Knock on wood. But I don't like the no voice remote that I have with the TV. So I don't know if I'm gonna keep the same TV. Okay, I'm flipping over here to target cause maybe we'll do this, we gotta get something. So I'll watch the video long, target short. The video doesn't look like it's ready. It's not ready yet to go. So stop's gonna be 133.50. Let's do 40 by 50. It's a really good entry. 40 by 50, short target. And I'll flip back around into video. Again, quick target on this is gonna be 132. 40 by 50 is valid. We're on target short now. I'll go look at the video in a minute. We turn away, we could miss this. If you look the other way, you miss the hawk. Oh, that's not ready yet. Let's go look at the video. That's not ready yet either. That's not ready. Okay, not in anything yet waiting. Just gotta wait it out. So stop's still 60. We could do, we could do 60 by 60 target short. 60 by 60. New numbers here people, just roll with it here. 60 by 60 is valid. Target, target 132, 132, and see where we go. Here, you could do it. You could do it right here. Now let me just look quick at the landscape. Say exactly, this is, it's just kind of crazy. So what's going on this week? Let's review. Here, the video if you want it. The video if you want it, I put the shop at 428 and go long right here. It's not a bad entry. I'm just gonna do one thing. Here we go, we're almost there. Target, target, target. Again, fast trades and out, fast trades and out. That's what we're gonna do all summer. And every single day until earnings season starts, which isn't until July. So again, not till we get back from the July 4th holiday, which is a weird holiday because it's in the middle of the week. Here we go, 132 and a flush. Here we go. Yay, boom, ba-do, ba-do, do-do. 132 and one more push and we will be out. Anyways, the Fed is talking on Capitol Hill this week. Tuesday, no, Wednesday and Thursday. Wednesday and Thursday. So the Fed's gonna be on tap this week, even still. Here we go. You can get out of it right here or you can wait to see if it breaks. Hand of the button here, target. You can be out. One more push is all I want. 89, did anybody get out? Shit, I said 132 and then I, it broke and then I didn't get out. Is anyone out? Do-do-do-do-do-do, do-do-do-do-do-do-do. Jackie's still in it. Anybody else? Oh, it's starting in two days. Cloudy today in New York. I gotta go out for a park walk today because in two days, starting Thursday, we're getting rain every single day for a solid week. We need it. We need it. We need the rain. We're sort of like almost in a drought plus the smoke has gotta get past us. We need the rain. Jerk's out, anybody else? Anybody else, anybody else out of target? Where did the hawks go when it rains? That's a trivia question for the birders. Here, one more push. Woo, did you see that? One more push. Here we go, one more push. Target, push, push, push. Again, you could be out. So we're out. If we're still in it, hand of the button here for one more push in it. Then we'll see where the market's at, and the video and everything else. Now, if you're nervous about it, put the stop at break even. Put the stop at break even. If you're nervous about it, first of all, you could get out. You're up right here if you wanna get out. If you're really nervous about it, put the stop at break even. There's two choices. Just get out, A. B, put the stop at break even. Don't stress yourself out. Got enough stress in this world. My clock is definitely not right. The clock is off today here. Back, I should go up and see if I can see the hawk today. So I took my father and they were doing work on the roof. So he wasn't there. He was probably bothered by the people and the noise. So I took my father Friday and he wasn't there because there were people in the roof. Here, come on. 99. See how tricky this market is. That was Tesla. Nope, I didn't get out of it. I'm still in it. Who is still in target with me? Again, I said fast trades and I didn't listen to myself. I said 132 and then for some reason I thought I was gonna keep going. It broke 11 cents and I didn't get out. I hope this isn't it. I don't think this is it, but now I gotta wait. Again, fast trades, fast trades. I'm gonna set a buzzer. I'm gonna set an alarm every minute. Every minute on the minute. Tom got out. I think that was good for you. Every 60 seconds I'm gonna set my phone alarm. We're done here. We will go over the market. Anyway, so Wednesday and Thursday could be wild for the fed. Here we go. Here, done. He's a little bit more out of it. Again, this is lower. This is lower. This looks good, but I think we gotta get out of the day trades fast. All right, let's go over everything. Let's go over everything.


Hello everyone and welcome to Tomorrow Space Orbit 12.05. We're very excited for you to be here today because we have a very cool guest with Joe Bernard from BPS Space showing you some of the wild things that he is doing in the world of amateur rocketry. So once again this is Tomorrow Orbit 12.05. And I am Jared, head the host for today. You probably know me as the resident astronomer of Tomorrow and right next to me I've got Sarah Vincent who what are we calling you today? In a certain nickname here, you pick. Okay, yeah. What do I do? What am I? Yes, what should I be? We can bring it up to a community folks. Yeah, I'm up for that. And we do most everything with community folks. Sarah McSaraface would be hilarious. Okay. I'm just putting that out there. Sounds good. Sounds like a good new Twitter handle if you need it. Yeah. And then we have Athena. Hello. Athena, how the heck do you say your last name? Brentsburger. Brentsburger. So Athena Brentsburger and you are like our resident super like super stellar explosion expert here. Yes. So excellent stuff. And then all the way on the end, as we mentioned in our intro, we have Joe Bernard from BPS space today. And we're very excited to have you on the show here. I know I'm excited because I'm an amateur rocketeer myself and some of us here are amateur rocketeers. So I guess just to kind of start out a little bit, just kind of give us a little bit of background about yourself and maybe how you kind of got involved in rocketry. Yeah. So my name is Joe Bernard. Like you said, I run this company called BPS.space or BPS space. I don't really know like what to call it. That's the website though. And also the name. Anyway, we build these really advanced metal rocketry components. And I kind of just cut like I want to see how crazy advanced you can make these small scale rockets. A lot of times when rocketeers want to build like bigger things, you're an amateur rocketeer. Yes, bigger is better. Well, yeah. So that's like the common view that like a lot of people take. And actually, in a lot of ways, bigger is better. But you can also get really advanced at the model scale. If you start incorporating like guidance systems and things like that into your low power rockets, you can stay really safe and still learn a ton about how the real things get to orbit or get to wherever. So that's kind of what we do. I'm all self taught in it. So it's kind of also a journey of like, how much can I not mess up on like my path through learning this? But yeah, that's about it. I studied music. And then I saw what SpaceX was doing in 2015. And it was like, uh-oh, I picked the wrong career. Yeah.  So you literally just like dive in the textbooks and that's how you learn physics behind all of this. It's also crazy. Like it sounds like a joke, but for real, there are a lot of YouTube tutorials that like work pretty well for real. Yeah. Excellent. So with this, did you also have to learn how to kind of like work with the hardware that came with it as well? Yep. So I'm not a mechanical engineer. I mean, at this point maybe, but like, I wasn't. And so I bought a 3D printer. I got like some CAD software. It's like, you know, computer aided design. And again, like YouTube tutorials, get you started pretty well.  You can iterate so many times, like, you know, print seven versions of the same part in the same day. And basically just like make it a little better each time. So it's really forgiving if you're not a good engineer to start. Yeah. Yeah. 3D printing, you know, rapid prototyping. That's like, you could do tweaks on the fly if you need to. Yeah. And we see this in like all of the real aerospace companies, too. They're moving all of their stuff to Incanel or whatever it is that they're 3D printing for some of the more complex parts or just to do rapid iteration on their designs. So when you got into rocketry, were you really like looking to push the boundaries of what amateur rocketry could do?  No, definitely, definitely not. The way that it started is so I mentioned I saw this, I saw this video, it was specifically the F9 DevR test. One of the ones in McGregor goes up to like, you know, 1000 meters, comes back down. And I saw a video of this on Facebook. I was like, uh oh, that yeah, that's the uh oh, I picked the wrong career moment. I kind of figured, like, I wanted a job with these people. And I knew that, like, I couldn't show up with a music degree and ask for a job. Like, I wouldn't be taken seriously and rightfully so. So I had to demonstrate it in some other way, or I had to go back to engineering school. And that second option is really expensive, takes a long time. And it was a literal shower idea where I thought maybe, like, I knew that Arduino existed, that all of this like, open source coding existed. I knew that you could get into 3D printing and like, print a bunch of parts. There were lots of different rocket engines and rocket parts available. And I figured, if I worked hard enough, I could probably like, try to launch and land a Falcon 9 replica. And that would be enough where I could like, tweet it. I didn't really think of the logistics here. But like, I could like, tweet it to SpaceX or something and see if they would say, hey, do you want an interview? So that's how it got started. It's mostly I just, I just wanted a job in the aerospace industry. And it's something much different now. It's, I just like, along the way I've realized, I just find this stuff so fulfilling. And I love sharing what I've learned online, even when it doesn't work, like you end up getting a lot of good feedback and becoming a better engineer from it. But that's what I do. Yeah. Yeah. That's so cool. We actually have a question from the chat. It's from Gregorius Soda-Harmel from YouTube. And they ask, what is the limit on guidance before it became illegal? Man, that's a, that's a good question. There's sort of the joke in the amateur rocketry community that, you know, you really can't put guidance on it because once you put guidance on a rocket, it becomes a missile. And the United States government isn't a big fan of people who aren't the United States government getting missiles. Yeah. I honestly, I don't know why. Because it's like, you deal with like legalities, but I love talking about this. So they're basically, there are like three regulations that you want to pay attention to in rocketry. The first one is FAA Part 101, which deals with model rockets and that different, there are different classifications of vehicles. And none of it mentions active guidance or active control. The second one is the National Association of Model Rocketry as a safety code. Again, there's no mention of active guidance or control. And the third one is another organization, Tripoli, and they don't mention active guidance or control anyway. So there aren't actually, I have to be careful about this. There aren't actually, like the part that you get, because you can get in trouble for this stuff, but the way that you get in trouble is by, usually by exporting it or sharing enough of your information. And so actually you see a lot of university teams with really high powered vehicles steering away from this stuff. And usually for good reason, because export controls regulations, this is like ITAR or the USML, these things deal with the sharing of information as well as the sharing of hardware or software. So what's USML? USML is basically the same as ITAR. That's the United States Munitions List. Okay. If you want to read it, it's, it's like, I don't know, several hundred pages of just really confusing words. But yeah, so it is, it's generally, and then the other thing is there is some gray area in this too. But if, I think generally if you're doing it with good intent, and if you're not strapping any type of, oh, oh, I didn't differentiate guidance, by the way. So my rockets have stability and not guidance. And the difference between the two is stability just tries to keep the rocket upright. It's doing the exact same job as a fins on, as a set of fins on model rockets would, right? So you're just trying to make sure it stays straight up in the air. Right. And guidance would be maneuvering in reference to some point in the real world. So like you're trying to, I mean, you shouldn't do this, but like you're trying to hit a point or you're trying to land on a point, you know, like if you're landing a model rocket, but mine just have stability. So they try to maintain upright, maintain stability. Oh, okay. Yeah. Got it. Okay. So stability is a lot more safe, legality wise than guidances. And I think things get a little bit more tricky there. Yeah, because then it's on the lines of possibly being more like a missile if it actually has guidance. Right. That's when it can direct, like you were saying. Yeah, I think there may be like, I think there's some, there's some amount of scale that gets into this too. If you build something that's guided that weighs, I don't know, like a half a kilogram, it's probably not going to be an issue. If you're building at like something that's like 20 kilograms, you're going to probably get some calls about that. Got it. Yeah, just a little bit. Yeah. Because yeah, 20 kilograms moving really fast is quite energetic. Yeah. Yeah. It's tough, though. There's, there is some gray area in it. I just, I generally speaking, though, there's actually a lot of information about this on the BPS.space website and the about section. I have lots of, there's resources for like the legality around this stuff. But yeah, you generally want to stick to stability instead of guidance. Gotcha. Wow. So what were those first, like, what was, what were you doing at first with your rockets to develop the stability with them? Like the first test flights? Yeah. What was that like?  So what, like, what's happening during this early timeframe with your flights? Yeah. So the schedule, the schedule for most of this stuff is like, you wake up, you go to Starbucks for three hours, read rocket propulsion elements by George Sutton. This is like, this is like the the Bible of aerospace engineering. But yeah, so it's just a lot of that, then you go home and I basically built like a rigid schedule of like, making sure I was focusing on the, on all of the different aspects of this. So you have like mechanical engineering, you have electronic, or electrical engineering, that's like your wiring, you have software design, things like that. But these first test flights, the first test flights were like awful. The first one, the first one was like a really obvious error that if you, if you showed it to anyone without any further results, they'd be like, yeah, you're never going to make this work. Like, if you, if you did, if you made an error like that silly, it just won't work at all. And actually, that was like the first 10 flights or so were pretty bad. There was like one that was really lucky to have looked really good on camera. But it was for real, just because of like the angle of the camera. Yeah, I mean, they were all bad. And so you learn a little bit more every time you look at the flight data, you look at actually one of the benefits here. So I mentioned off air, but on air, basically, I was a videographer, as I was learning these things. So especially with shooting weddings, you work a lot on the weekend. And then during the week, you just spend a lot of time at Starbucks reading books. But yeah, because I was a videographer, I had access to all of these nice cameras. There was like the company I worked for had a red camera, and we were able to shoot a lot of high speed footage to, so that I could quickly diagnose after a test flight, what was actually going on on the vehicle in addition to the flight data. So this is again, this is like 3D printing, it's the iterative design. If you can afford to just go through a little bit of frustration, it's a really fast way to like get things moving. Yeah, and was there a certain like kind of regime with your flight testing that you were working within? It changed a few times. The first few, okay, yeah, yeah, it changed a few times because the first series of vehicles were called the scout rocket. They were actually dog themed. The scout rocket, it's like scout's kind of a dog name. The computer's name was Fetch. The launchpad software was Throw.  Because I mean, your company now provides what other people can buy, but you didn't have access to you. Yeah, no, I did not have access to Future Joe. I wish I did. It would have been a lot easier. But yeah, so you go through a lot of like a lot of garbage designs, a lot of silly errors. You end up with a lot of those too. You neglect to check a wire that's stuck in some mechanical connection. Always, yeah.  So yeah, the first, you asked about the flight objectives or like the parameters that you're working with. So the first few flights were of the scout vehicle, like I mentioned, and the goal was just to launch and land in the same flight all up with no experience doing it, which as you can imagine doesn't really work so well and you end up smashing a lot of rockets that you've put a lot of work into. So after about a year of that, I had spent enough money and enough time where I was like, this is not working. We need to switch to something different. So I built a new rocket called Echo and a lot of these are just aesthetic changes. They use a lot of the same flight computers and components. But the new rocket was like the airframe was much more stable. I basically built it so that if it hit the ground like really hard, if it didn't deploy any parachutes and just free fall right into the ground, it would still be okay to fly in like a day or two. And that helped a lot. That helped like really speed up the time between launches. So Echo was like a new, not like mission objective, but like test profile, I guess. So reusability is key. Reusability is key. It is. I love how much experimentation was done and that you do mention that a lot. And I think actually on your website, there's a line that's my favorite talks about how experimentation is really the way to, you know, start to achieve these things. So I have a really awesome question from the chat, from Jake and from YouTube. And they ask about actually the fact that you have kits. They say, are we going to ever do model rocket competitions in school? And what do you think? What type should we build? Water rocket or solid fuel? And I think that's super important that your kids bring experimentations to people. So what are your thoughts on that? Yeah. I mean, definitely, I think, I think model rocket competitions are awesome. I actually, I've had it in my mind to like start one a couple of times, but it's like a non-trivial amount of work. Yeah. But I think the other thing I mentioned on the website, whether I mentioned it or not, this is just how I feel about it. But like experimentation is the way that I learned best. I don't think everyone does, but just being able to have hands-on tools like a regular model rocket, a water rocket doesn't even matter. Something that you can actually play with or work with to help you gain an understanding of a new concept or of a new thing that you're learning is just crazy helpful. At least for me, I found that the only way I'm able to really learn something, like until I actually interact with it in the physical world, I haven't really understood what's going on. Right. Exactly. So I think competitions like that or just any incentive to get kids to like play with model rockets or play with any engineering thing. Like I did first robotics in high school, massive robotics competition, and it for myself and for all of my friends who were involved, it was just a huge, it's like so important to fostering that like innate desire that a lot of us have to build things or to experiment. Yeah. Well, speaking actually of experiments, really another good question from the chat. I love this one. It's from WKD from YouTube. It says, with the amount of launch experiments you do, which bums you out more? Bad launch or not getting the shot? You know, I think at this point it's a bad launch that disappoints me more, but that was only like a really recent flip because at this point there's just a ton of media that I can pull from to be like, look how well things work, even if they don't some of the time. But yeah, because I have a videography background, I also really want to get the shot. Sometimes I'll do a launch and my shutter speed on my camera will be too low and they're all blurry. They're all blurry or something like that, but at this point I think the engineering is more what disappoints me than the actual, as long as I get the flight data back it's fine. Got it. So what began to happen when you started having success with your rockets actually coming back? It felt really good. It was like a solid year of failure, just like one after the other where each time it was more frustrating and then at some point the rocket went up and it didn't like fishtail everywhere and it didn't flop about. It did actually fail to deploy the parachutes and slam it to the ground, but the up part worked. It's kind of how we watch SpaceX go from not being able to land anything to as soon as they could land it once and get all of that data back on what a successful flight looked like, it just got a lot faster. It just really ramped up the pace and it's the same thing here is as soon as I knew what worked I was able to really focus on the things that were working and what still wasn't working. So everything just accelerated at that point. Yeah, so verification was really critical for moving this forward. Yeah, for sure. And even, you know, it doesn't mean that at that point things were perfect. Like every single part of the rocket has fundamentally changed since that point. We went through like a period of like not launching anything over the summer of 2017 where I was just revamping all of the math behind like the flight software and flight simulation and things like that and that really improved reliability too. But, you know, as soon as you get, if you can get like that one success, if you can make it to that point, usually you can make it a little further. Yeah, so speaking actually of moving forward, some pretty good questions in this chat. So I'm going to specifically about that. They're called Apogesis. That's pretty good. Oh, such a good name. Remind me to yell that out. Apogesis. So they ask, how do you see your development pathway going forward? Better usability, switch from solids to pressure-fed liquids, advanced multi-stage? Yeah. There's so many cool things in directions that this could go. So I think we've got like three major things going on right now. The first is this series of videos that I'm producing. It's sort of educational and it's like, if you were going to start from nothing, if you were going to start from where I was in 2015, here's exactly how you'd go about landing a model rocket. Now, I haven't done it yet, you don't have to follow the series and believe what I say, but I feel like I'm pretty close. And that leads to the second one. So the first major one is producing that video series. The second one is actually landing the rocket. We've come really close. It's like two meters off the ground and it hovers and it just, it's so close. I'm in the middle of a massive software update for the landing stuff. We're going to stick it in 2019 for sure. But you heard it here. Yeah. Yeah. You heard it here. On tomorrow, 12.05. Yeah. Yeah. On February 2nd, 2019. You heard it here. I'll stand by it. So that, and then the third one is the Falcon Heavy model. I built a 148 scale model of the Falcon Heavy. Yes, that was awesome. It's like, look at it. It's so, look at how beautiful it is. It's so true. Exactly what Elon said about it, which is that on the surface, it's like, cool, you strap three rockets together. Except it's like a billion times harder. There's so many, there's so many weird considerations that happen when you put three of these things together. Anyway, those are like the three major projects that are, like I'm working on now. And at some point, the landing is going to sort of wrap up because we'll have done it. I mean, that's hopefully, I suppose it couldn't happen. But yeah. Yeah. There's some bigger projects that are coming up in like later 2019. I don't know how much I want to say about it. Okay. That's her. The rockets are going to get larger. That's so exciting. Really good question from Minnie Stoge. Actually, hi Stoge. And she asks, are you going to start building your models out of carbon fiber, like electron? Yeah. Speaking of future stuff. That's, it's such a cool looking vehicle. Yes, it is. But yeah, so actually, where does small enough scale, at least with these rockets? Most of them are about a meter tall, right? They're not, they're not that large. And so it's actually, you know, as long as you're not hitting the ground that fast, it's better to just build them out of cardboard. That's the lightest material. But like some of those larger projects, actually, I can talk about one of the larger projects. Okay. We're, I'm building a reaction control system for a couple of high power rockets. I'm working with a buddy of mine up in Boston with it. But yeah, that's going to need to be built out of a better material than cardboard. Because that's going to be going pretty high and really fast. Got you. Yeah. Yeah. I mean, if you're going high enough to need a reaction control system, holy moly. I kind of want to ask you about the challenge of using solid motors, because I think you're just using only solid motors at this time. And that, you know, you light it and it's lit and it's burning until it runs out of fuel. And there's no change in it. Yes. You do have a mention of you can throttle back a little bit if you do certain things. Yeah. Okay. So we'll, first we'll address like the solid motor thing. Basically, it's a little bit cheating, honestly. If you know your flight profile well enough, like ahead, if you know that you're going to, I don't know, have an apogee between 50 and 60 meters or something, just as an example, you can get really close if you spec the motor that you use to land. Even if, even if you don't, even if you don't get it quite right, like the legs should be able to take a little bit of the impact if it drops. And then if it hops a little bit, hopefully you can just maintain stability long enough that it comes back down to the ground. So it is cheating a little bit because I know exactly how high the rocket is going to go. And is that cheating or is that just really good science? It's just, it's just a lot of simple, it's simplifying a lot of the problems that are with landing. But I think that's kind of the key of engineering, right? Yeah. Know what you need to do and get it right. Yeah, whatever accomplishes the task.  Yes. But okay, so then the second part is, you mentioned the throttling, the solid motors. That can actually be done. There are some interesting ways that you can do that. You could do one, you could do like a system where you, this is, this is, I refer to it as like cosine throttling because you get cosine losses in your motor. So you have like two of them pointed outward and then you sort of gimbal them outward as you want less and less thrust. That's one way to do it. You could stick something in the exhaust plume that like, you know, takes away some of the thrust. You could, there may actually be an instance of this later this year, but you could perhaps change the size of your chamber. That sounds like the hardest possible thing to do, but we'll see. Yeah, yeah. Because I know a shuttle, with a shuttle solid rocket boosters, they would basically change the surface area that would be burning inside of it. Right. And that would allow the SRBs to actually be able to be throttle-able if you wanted to with that. Yeah. And the reason, you know, one of the direct things, if you know what like a hybrid motor is, a hybrid motor is sort of a bridge between a liquid and a solid motor. They're scary. They're scary. They've got all these problems, but they're like, you could certainly do this with a small liquid motor or a hybrid motor, but you know, there are also, there's like a certain amount of challenges that I'm willing to accept in a certain amount where I'm like, I'm just not going to touch that for a little while. So hybrids and liquids, like you're just not interested in the moment. Well, not right now. Okay. Got you. Yeah. There's, there's bigger projects coming, but not right now. Got you. Which path are you leaning toward? The gimbling out, or are you thinking maybe compressing the... I still think it can just work with one. The motors that I'm using, like in model rocket motors, you generally expect like a 5% total impulse variance. So like, how much, how much total energy is your motor going to give the rocket? And there's usually about a 5% variation. But I've done a bunch of testing on the specific motors I'm using and it's like, I found it to be within like one or two percent. So I think that's enough that if I can get an accurate enough start on the retro burn, I think I can reliably touch down. Again, like we'll see. Who knows. But, and if it doesn't work, I think the right way to do it, I outlined this in one of my videos, but I think the right way to do it might be to, well, the one that I would choose is you have a small set of tiny solid motors on the outside and then you sort of underspec the main landing motor so that you can sort of do, it's almost like, you know how the Soyuz lands or the Blue Origin New Shepard cabinet, you can just push it down. So you do like these little impulse motors on the side that just add or subtract, well not subtract, but like add a little bit of energy. We'll see. I don't know. Okay. I have got a question for you. Actually, as you're speaking, I'm thinking a lot about, okay, what can be like this used for eventually, because it's awesome. You know, it's really fun. But, and then you made these kits and I'm like, what else could it be used for? Well, there's this awesome question from the chat that addresses this. It's from AL13N and they actually ask, if you propulsively land, can you actually put payload on it? So is that a thing you thought of maybe doing as like doing some type of payload and eventually turning this into something you can collaborate with other companies on? Yeah, you could certainly do that. I think that's probably not the path that I'd take with it. I think if you, like the height that most of my rockets end up going to, it's just going to be more cost effective to send it up on a drone.  But sometimes the comments are like, why, you know, why would you spend so much time on this? Like there's no real use for it. And I actually, I have to agree.  It's just like, this is a cool project. I like doing this stuff and I like teaching people along the way. So I think, you know, maybe there are some commercial applications for it, but mostly I'm focused on the engineering and like how cool that is. And the educational aspect too is huge. I mean, the fact that I was watching one of the videos on your website and, you know, you're saying as soon as the kits hit the market, and you weren't even going to do kits at first. There's people recommending it. And as soon as it hit the market, like sold out right away. And that's just, that's extraordinary because it's starting to then inspire the future generations to want to go into this. And it also is worth mentioning, like you'll never beat like the simplicity in model rockets of fins. The point, yeah, the point isn't that it's useful really. It's that it's cool. Yes. Yeah. Anyway. Which is, which is a perfect reason to do it. I mean, you know, I've seen people turn things that shouldn't be rockets into rockets because it looks cool. Yeah. So I mean, who doesn't want to see a traffic cone suddenly flying? Yeah, a Tiki bar. I mean, speaking of cool things, like I do want to actually talk a little bit about your music. And then I realized, I was like, that's something I definitely wanted to bring up. And there is also a comment from the chat. Also awesome chat room. It comes from WKD. And they off of YouTube. And they asked, what's the deal with wedding photographers and space enthusiasm? Who knows, man? Like, what is that? I guess it's a thing. I don't know. Maybe a lot of other people out there are wedding photographers and love space travel. But but we're having this, this happened for you. I mean, yeah, like I know three off the bat, like Brady Keniston is a wedding photographer. Really? Tim Dodd was a wedding photographer. Tim Dodd was a wedding photographer. Yeah. What's the deal? What's going on? I believe he's watching too, if you want to say hi. Tim Dodd, the everyday astronaut. Hello. Yeah, I don't know what that is. I like have no answer. I will say that like, I really didn't enjoy being a wedding videographer. A lot of people do. Does anyone? I mean, I'm not going to speak for everyone. The same music. I didn't enjoy it more than any other job I've ever had. Like, I didn't enjoy that so much that when you do a job that you don't like, your mind goes immediately to what you want to be doing. Yeah. And so it was as soon as I started doing wedding videography and I saw the SpaceX stuff, I was like, cool. Like there's a there's like a countdown on me doing this job. Yeah. That's so epic. Do you do anything with music still? Yeah. So I studied at the Berkeley College of Music. I have this really nice education in music. And I kind of use it. I mean mathematics, music theory. Well, yeah, I mean, yeah, to be fair, there's like, I did a lot of acoustics stuff at Berkeley too. And there's like a lot of cool math and like determining the mode of a room or things like that. But I do music kind of like Tim Dodd. I make all the music for the videos that I put out. It's not a ton of stuff, but yeah, it's fun. Music is great. I kind of want to ask you too, as someone like myself who started like in an art background and is now going into engineering and things like that, what was kind of the have you gotten a reaction from the aerospace community on this and kind of like, how's that been? Because I know that when I went into the aerospace community, it was kind of like feeling like an outsider coming in initially. Like I was someone with a different perspective than everybody else coming in and kind of like, I don't want to say being shunned, but not necessarily. It doesn't feel good. Yeah, it doesn't really feel that great to be around initially. Dismissal. So yeah, a little bit of dismissal. I think that's normal. I mean, I felt that for a while. There are still situations. I presented some of my work at the Marshall Space Flight Center in Huntsville, Alabama. And I remember, so I mean, they're all brilliant in ways that I just will never understand. They're all just brilliant down there. And so I presented my work and naturally as engineers, they want to know the details. And there's a real limit to my knowledge on this stuff because a lot of it is learned by experimentation, which means I have these weird holes in what I know. So they would ask a question that to them seems really simple and I'd be like, honestly, I don't know. Right. So there's still stuff like that. I think, I don't know, I think everyone feels like an outsider at some point. Do you have any plans to maybe get, you know, part-time work on an engineering degree, maybe having the math and the actual materials knowledge that they're asking about? Would that give you kind of an edge on a little boost for what you're doing?  And I think it honestly is a good thing to do. I don't have any hard plans for it right now. But I think, especially if the ambition is to work in an actual aerospace job, which right now, it's just a further BPS. But if that's, if my goal switches back to like, I want to work for an aerospace engineering company, I think the responsible thing to do is like to go back and get your degree. There's a, there's like, I think there was a little mini doc that motherboard did just recently. But I mentioned it in this, and I'll mention it again. There are some things that work for self-teaching, like for software engineering, where you can mess things up a bunch of times and just re-upload your code. That's fine, I think.  Maybe not. And I think aerospace is somewhere in between there, especially if you're working at the larger scale, the stakes are really high. And you kind of want to really have that good base of knowledge. So for the model rocket scale, it's fine. But at some point, you know, you probably want to get the degree. And when you do these test flights, you kind of have like an audience with you there watching them. Like, do you, do you do them at like little events that they have on the weekends or? I'm not, so I'm not superstitious, except for this one thing. Because as it goes, so.  It goes terribly wrong. It's like one time, one time I brought my parents and my grandfather and like my friends out to this test site and it was sunset and it was gorgeous. And the rocket goes up super straight and then it comes down and it's time to deploy the parachutes and it deploys them in the parachutes work, except that they're not connected to the rocket. Oops, oops. Yeah, indeed. Oh, man, indeed. And I completely smashed rocket one time. Actually, just recently, so my friend nanny is for this little kid.  And so I was like, hey, come see a rocket launch. And that whole week I tried every single day to launch the rocket and every time something would fail with the motor ignition or something. I made a whole video about it. I was watching that before. Yeah, failure and burnout. So great. And it's just like, I don't know what it is. I wish it wouldn't be the case. But honestly, like half the reason I generally just have myself and no one else at the launch site is because it's not a safety thing. It's just like things seem to just go wrong when other people are there. Performance anxiety. Yeah, that's what it is. So no possibility for a live stream is a question we have from the chat from PCS Locked. What if Joe Livestream's next BPS space launch and Tim covers it just as a real rocket launch? The live stream is a different thing. I would be down to have someone else live stream it. Okay. Because the live stream is mostly just I'm focused all on getting the rocket set up and then getting my camera set up. I don't know, maybe like all of these other little tiny things that have to happen at the launch site. And a lot of my energy can't be focused on like, what's the chat doing right now? Right. So if someone else came down on live stream, that would be cool. But I'm probably not going to do it just myself. Yes, you're willing to show it live and not talk about it like three days later like Blue Orange or something. There's a lot of secretive stuff that they do. Yeah. And then it won't be superstitious anymore. So yeah, there we go. You'll break the superstition in my mind. Yeah. Oh, and we actually have a question from Tim Dodd the Everyday Astronaut. So he asks, how do you land with the solids? One motor, two motors? How does the computer calculate when you ignite the next engine? Cool. Yeah. So hi, Tim. What's up? But yeah, the way that I do it, it's sort of like brute force through a lot of simulation. Actually, you hear Elon Musk talk about this sometime called a Monte Carlo simulation stuff. So for calculating when you want to land the rocket motor, I have a flight simulator offline that I just I run a ton of simulations and find like, cool, with this given profile, with the vehicle mass, with these two, it's two motors, you have one motor for a cent, and then it burns out. And they're actually, I think I just, I talk about it a little bit in the Falcon Heavy video, but they're actually stacked on top of each other. And the second motor literally hot stages the first motor out. Nice. When it's time to land. It's like an old school atlas. Yeah. But it's a, yeah, so it's sort of brute force through a lot of simulation, you determine what, where is like the range of altitudes that I should be burning that retro motor. And then it's pretty simple in the flight software. It's nothing like the real Falcon 9 uses this super complex thing called g-fold. But my rockets mostly just look at the apogee, they look at a couple other things like the orientation, if it's going to be safe to burn the motor, things like that. And then they just sort of call it when they see it. Nice. Nice. Yeah. Very cool. This is super cool. So overall, what has sort of like been the biggest takeaway from all of this for you? Like what's the thing that's affected you the most that you would kind of want to throw out there for everyone to take? I think this might seem obvious at this point. So BPS has a, has a decent size falling at this point, but share your work online. Like, even if it's, even if it's not good, especially like you can see, if you go all the way back to like the beginning videos on the BPS YouTube channel, or like, I don't know, it's really hard to go all the way back on Twitter, but whatever. All the way back, like even if, even if stuff isn't working, you should still share what you're working on online. You don't have to share all of it. You don't have like open source it, but if you share it, you'll become a better engineer because of it. I definitely have, like every time something has failed, sometimes it's frustrating when someone's like, well, why don't you just do this? But a lot of the times people just want to help and they just want, and you know, they have different perspectives that will help you. So I guess the takeaway for me is like, make sure, like if you're doing something cool, tell other people about it. Don't, there's a lot of excuses you can make to not share your work online. And I think most of them aren't, aren't really valid. Tell us a little bit about BPS space. What would you like to know? It's mostly, I don't really understand it still. So is it like a resource for people to go to? They could buy, they could buy kits as well from you? This is hard. It's hard to describe because I still don't really know like what to focus on, or rather like what to focus on when I'm talking about it. But it's a couple of different things. I think I'd like to make it more of a resource, which is why I'm doing this like landing monorockets video series. So right now we're in the middle of designing a new computer with all of the, you know, specific things that we want to land the monorocket. And then after that, like I'm just going to bring everyone along as we design all the mechanical parts and things like that. But yeah, so it's a little bit of a resource. We're also a direct to consumer. Like we sell these flight computers for Thrust Factor Control and monorockets. They are out of stock right now. I'm working really hard to fix manufacturing stuff. But yeah, we sell those. And then a lot of the, you know, some of the bigger things are, we release a lot of the files that come out with the landing monorocket series on Patreon. So we're, a lot of the reason that this stuff still exists is through Patreon, which is crazy. It's amazing like the generosity of some people. Yeah. Yeah. And if people would like to get more information about you, Joe, where can they go to? Literally, bps.space. You don't need the www either. Got rid of that. Yeah, bps.space. That's the website. All right. Well, Joe, thank you so much for coming on the show today. Thanks for having me. You're doing some amazing work in an area that has really like not seen much advancement for a very long period of time. Monorocketry has been pretty static in terms of what it can do recently with like Arduino and everything. It's kind of come to a point where you can like do some really cool things, but you're like, you're making it exponentially cooler with being able to land and I'm really looking forward to the big things because I would like to land my big things with that mostly because I always have problems packing my parachutes. So, Joe, thank you so much for coming on the show today. And of course we want to thank you, our viewers and our patrons as well, our escape velocity citizens. These folks give us $10 or more per episode to help us out here at tomorrow. Also, our orbital citizens who give us $5 or more per episode as well. And then, of course, we have our suborbital citizens. You give us $250 or more per episode and we'll give you a little bit of time to find your name here. And of course we also have our ground support citizens as well. You can attempt to find your name there as best you can, very itty bitty font. And I love the fact that that font has to get smaller and smaller as things are going on. And when you support us here at tomorrow, you're able to provide things like these amazing interviews like we got to do with Joe today to talk about BPS space. Also, our new show, which is brand new, which everyone seems to like and we're going to definitely keep doing that as well. And maybe some things in the future as well with that. And of course, we do give you rewards back. You can check us out on Discord at our Discord channel. You can also come on over to community.tmro.tv as well because it's not just financial support that we welcome. If you have any sort of skills that you would like to help us out with, you are more than welcome to bring them to the show. And we will utilize them and we'll put you to work if you want to get put to work. So on next week's show, 12.06, we are going to have Asgardia come on in to talk about the attempt or I guess they sort of are a space nation. Yeah, I know. I think me and Space Mike actually signed up to be a part of that. Oh, and Ben too. So I don't think we've paid our citizenship fees. Oh, you've paid yours? Okay, I haven't paid mine. So I guess it makes a mare or something. Are you like the overlord of Asgardia or the overlord? He said you're the lackey. Okay. All right. So thank you so much for watching tomorrow orbit 12.05 and we will see you next time.