 Good. Okay, so let's I want to say fairly briefly, how to make things make sense, because I feel like the making things make sense is the is what's important is that you can make things make sense not how exactly you do so. And I'm not convinced this is the perfect way to make things make sense they're just like many ways of making your definitions work. So, but just to make this sound. I want to show that the arguments are kind of robust and really what this is saying is that Bob period is very robust that it works that. So, so the, I'm going to describe a setting in which our arguments work over any reasonable base. So over the complex numbers if we want to talk about topology but it can work over any field or even over the integers, and we'll be able to base to change our base, and we can have hodge theory homotopy types a tall homotopy types working for the ring. So if you don't know what these words are it doesn't matter in place you just these standard things you want to potentially have you can have they're just going to they're going to be kept on for the ride. And one thing we will need. So I guess those are the things that are added bonuses, and then in the course of the argument will need to have things like stacks and colon that's. Again, that's I already told you not to worry about, and then you'll see we'll have to worry too much about that. So, okay, so here's what here's the here's how you here's one way of making things you can work with. And if you are a fetishist for infinity categories there's an infinity category version to. So you just start with the reasonable spaces we're thinking about. You may disagree that these reasonable spaces but say smooth irreducible nice art and stacks, and I think everyone agrees that these things are all nice and then I'll like forget. Don't worry about that, but start with reasonable spaces they reasonable geometric spaces, and then what I want to do is define when a map topology like map where I want to say what a map is. I want to be connected and apology language I want to say when it's basically an isomorphism up to high codimension. So let me define what I mean by that. So I'll call this class of morphisms iso codimension k things that are isomorphic, essentially isomorphic up to codimension k. The first example I'll give you in the definition will show you how it's slightly unusual which is I want to vector bundle effects to V to y is a vector bundle. I want to call that nice morphism because as far as topology is concerned, that should be if you're over it that should be an isomorphism or if you're doing with Chow groups or any of the sorts of things we talked about. It is basically nice morph. It's not really nice morph. I'm going to want it to be nice morphs from topological. And in the course of our argument, we had to be able to take affine bundles and so that's so so I want that to be an isomorphism and any hood dimension, and the section of the vector bundle should also be something which should be considered, essentially an isomorphism can point to any of those sorts of things we've been talking about. And then finally, if something is an open embedding of something smooth and something smooth, except you throw it something of codimension k. I want that to be an isomorphism to go to mention k that's a very reasonable thing. So in terms of like the topology that they should look very similar in terms of homotopy groups if you know about things like that. Finally, in my definition, I'll just declare that a two out of three morphisms, if I have two morphisms, composing to give a third, if two of the three are in this class the third is as well. So I just make it a nice class and that's what it's a rigorous definition. And now I'll tell you what my cat new category is going to be. They're just going to be convergent sequences, by which I mean a sequence of maps of our nice spaces. So it's not them to converge in the sense that they are getting to be isomorphisms and higher and higher coding. So it's an infinite sequence. And they get to be ISO ISO code and higher and higher and higher as you can find. So that's that that's a reasonable thing. It's a code. So think of it as like a convergent sequence of our spaces. If you want to think like a topologist it's like converging to some topological space and you know what that is. So the morphism of the sequences where you might not be surprised that I want, I want maps, I'm not going to require that they go nice and horizontal, they can just like you might imagine with the convergent sequence all that matters is that the maps have to go kind of upwards. That's a map of Cauchy sequences with some equivalence relation is one caveat that these guys don't have to commute. They only have to commute up to a one. So if you if something is mapping, and you can slide it over by a homotopy by a one to get something else that should count as the same. So, so that's one caveat that will toss in. We saw that when we push forward things and we had, well, we actually did see that in the course of the proof. But again when it comes to any of the things we talked about that we wanted to have in our desiderata. So we just want maps of Cauchy sequences where the squares or whatever these rhombire trapezoids to be essentially commuting up to, up to these a. And that's it. One last thing I want to say is that I want to then say if I've got a map of these guys, and they're ISO, the isomorphism is going to mention K for all K. I want to call that an isomorphism, like a vector button because that's the last reasonable thing I want to have. So the definitions can try to make all the things we want to be true be true, and all the properties we wanted to have be happy. So that's basically it so it's a, so it's, but, and again, this is not cheating this is how you do science like you make your definitions after you, you figure out what's true, and then you figure out the words for what you want to say. And again, I will say explicitly, this does give a proof of original, like if you, if you take the original space, if you take the space and you take restrict to the complex numbers and take and litigation and take homotopy type that map is the bar period is to map this is a you get a that this does give up for bar period, but this is honestly an enrichment of that. So I will, so that ends my attempt to, that's the complex bar periodicity, and I now want to deal with the eight fold clock, which is going to get even more fun and the way in which it gets more fun is the way in which we guessed what that bot map was that gets more fun with chasing bundles around and it's things exactly what the topic of this seminar Lagrangian, it's, it's exactly the bread and butter of what, you know, what, what we have to think of. So let me stop for a second and ask if there are questions about complex what period is defined the map. We showed us an isomorphism by express by, and you even saw the entire basically the entire given all the details, and it's a philosophy of course. Right. So let me, let me now try to, so now I'm going to switch over to, to, to the period eight case. And what happened is I, well I gave a talk on the period two case, and there's this great paper from the previous monium of, of that in a previous lifetime when Jim was in a different field. When he was a topologist. He, he wrote, which looks like again it looked like algebraic geometry that really admitting to be algebraic geometry and it suggested this should be true. So here's what my curiosity is. Here's what the, here's the period eight case. So the. What it's all about is this eight fold thing where loops of each thing is the next one loops one of each thing the next one. I'm always talking about loops to. So what I mean, if you have questions after I'm happy to tell you why loops one is is is not at all surprising, at least half the time and I'm sure figure out the other half in the next little while to. So, so this eight fold thing where it's loops one, loops one, loops one, instead, what I like to focus on is the, the, this sort of interior square. And then if you understand the square you'll understand the others as well. And so here we're going to have maps of the sphere into, we're saying that the maps of P one and to be oh, should end up being the same thing as a point of this maps of P pointed P one of this point of this maps of point of this we pointed this and maps and P one of this she pointed this and that's what I'm about to try to explain. So let me start with this part of the square. And I should say this has got some sort of sort of rotational almost symmetry. So, I'm really going to, what I'm going to do will apply just as well to the part. So, what is SP module. Well, what do I mean by that well let me just focus on finite dimensional approximation, which is going to be SP module and module. So what this is is the, at least what I would call the isotropic Lagrangian gross money, you can let me know if I'm, if I'm saying this the wrong way so what it's doing is it's parameterizing. You have a two dimensional complex space, let's this over the complex numbers, and then abstract and say something over the general base. You have a two dimensional space. It's got this inner pairing it's got the symplectic pairing. So it's got a map to its, to its dual. It's got a map from to its dual, and it satisfies the fact that it's symplectic means this. And it's two dimensional. And what I want to do is have inside it an isotropic maximal dimensional subspace others an n dimensional subspace of V that pairs with itself, zero. So, now I want to map P one to this space. So what does that turn out to be what so so this is something which I think even some people in the audience have thought about in some ways, in very deep ways. In terms of quantum homology. Max was probably cross my knees and I have some, oh, I'm gonna have some. Is it back. Hey Robbie. Yeah, is my is it can you see my screen I should say personal. I cannot know. No, okay. I think to get that isotropic grass money and on the nose, you need to divide by the parabolic that contains GLN that GLN. In that case, I will say, so up to a one and whatnot, you know, that's nothing, but I think to say those words on the nose I think you want the parabolic not the not the GLN. That makes sense. And I'm not going to try to correct myself because I'll say something wrong. Great. So it's so it's almost but not quite this. And maybe more alarmingly you cannot see my screen. You can see your screens correct. Okay, so let's see what happened when I go to give me one second. Share content screen broadcast. But you know the robots at zoom recognize that you made that mistake with the parabolic and it just shut me down. I'll be kicked off Twitter as well after this. Okay, so it's not exactly absolutely I struck across money and I'm not even sure exactly how it's not now at this point but Jim knows how it's not, but roughly speaking it's got to be the following thing. I'm sure it's the following thing which is what so what what is loops to I'm now going to describe up to these a ones this is why I'm playing fast and loose with with vector spaces and a ones which are contractible what is loops to I'm going to put a bundle of SP mod you and I want to put something like this there and maybe this is going to be right. I'll get Jim is nodding. And so what happens I put my vector space V in the middle. It's a trivial vector bundle now. So this is a bundle. These are bundles on P1. I put a trivial bundle in the middle. One thing I learned. I did not know before is that I could I would just put a trivial bundle in the middle. I'll put a bundle and it's dual both of rank and just it's the same thing but it's and there's the there's the there's the inner product, there's the pairing right there. And so I'm parameterizing vector bundles, he such that they're dual. So that's such that you get this self dual exact sequence like that. So that's a map. That's what I'm parameter. And now when I parameterize this, I'm mapping like a P1, I think, into this nice projective variety this Lagrangian glass money. And it's got a degree. And that degree is, I'm going to say this right, the degree of the spectrum. And so what I want to do is let that degree go to an I'm going to claim that as this degree goes to infinity, this is going to stabilize or but differently, when the degree is low, I'm going to get some approximation to my final answer. What I'm going to do is I'm going to tell you, anytime I have a something like this, anytime I have an element of loops to SP module, I'm supposed to get a vector space with an orthogonal pair with an orthogonal pair. So where do I get that it's got to be some very simple recipe from bundles on P1. And this is something which I have expect, I have a 50% chance that someone's going to say, Oh yes. Of course, it's going to be this following thing. And there's a percent chance that Oh, I have no idea. So I don't know whether anyone's going to have seen this before. I don't, I'm not going to put you on spot. I'll show it to you and then you can see it. But I feel like I, I would not I only when you know to look for this would I have guessed it. So, great. So, so, let's see if I. Great. So, so, so, all right. So there are the maps to P1 SP module. So, so here's what I'll do. I'll, I'll twist this by minus one. And then I'll take the long exact sequence in cosmology. And now these guys are trivial bundles in the middle. So they've got no H zero or H one, because it's the trivial bundle twisted by all minus one on the one. And that goes to H one of a dual of minus one. And that goes to each one of the guy in the middle is also zero. But this guy by sero duality is a zero of the minus one. So here's my vector space, who is exactly the same vector space that I called a back when we talked about complex, but previously that's not that's a we have seen this vector space before, and we get a math aid a dual. And that's our map looking at it turns out. And that that is a that's a metric, not, not anti symmetric, we turn our anti symmetric thing to symmetric. That's that is that map that's our beta. And then how do you go in the other direction how do you the other, the other rotation omod you to be SP, it just works the same with the science change. And everything just works as is the science and change magical. So that's the, so that tells you this map, and this map. You don't have to worry about what magic this map gives you isomorphism. So you magically get this the other surprising map, and even better. This will give you guys. Okay, so now let me do the other. Finally, my final trick, I guess, is to get a map from P one. So I will tell you about this direction and equivalently this direction. So I'm going to get a map from P one to be oh, then I need to get a point of this. In other words, if I have a bundle on P one with an orthogonal structure, it's a bundle on P and it's rank n, and I'm, and I'm going to assume. So we're given a rank n bundle, and it's isomorphic to its dual by something where feed duals feed. That means I've given like a bump vector bundle P one and it's got this nice pairing over all of P one. So now find a point of a Lagrangian Ross money. Where will I find this Lagrangian Ross money. And again, I feel like this is being seen before, and I'm thinking about, in particular like that. There is this, this. I don't see the connection, but this will crash from back is two step trick that I just, but I don't see the connection. Here's what I'll do. So I will, I will tell you what the map is to an isotopic maximum less money. So the thing about E is it's isomorphic to its dual. So it can't be positive. It can't have only positive some ads when you take it splitting type. It's got to be symmetric. It's got to be like it's going to have some negatives and it could have some negatives and some positive, and only in the most generic case will they all be zero. So what I'm going to do is I'm going to restrict myself to open set just like we did in the initial popular Disney case. So it's restrict to a big open set where you have K is not negative. So let's just check the speeds. Well, we take the K of K plus one if K plus two and so forth. So if you case not negative. Now here's what I'm going to do. I take E of K minus one and you have to minus K minus one, and I take their, and I take the closure. So that's a vector bundle on P one. That's a vector bundle on P one. And this map is like you multiply by the coordinate at infinity. So I think it's supported, set theoretically at that one point at a one fixed point. And now once again, I take, what would you expect me to do, you're going to expect me to expect to see, long as I see this in cosmology, you'll see seriduality, and you'll see all the same pieces are going to be together in a different way. So when you take the long distance in cosmology will each not have this guy is zero, because of the numerology. And so I've each not have this guy goes to whatever this thing is goes to each one of the goes to each one of this thing, but each one of this thing will ease any dual or isomorphic. So by seriduality, that's H naught of E of K minus one dual. And there you go. And this thing is once again, our same a the thing that you take the bundle that was non negative twisted by minus one to each it's the same. It's the same space we've been seeing all the time. And now there we go we have a bundle. We have it's dual, and we have something in the middle. And what the sequence is soft tool. And so that is, that's telling us the map like this, and exactly the same thing with the science change to protect the innocent gives you the maps like this. And so what this, and then when asked to show that these, this gives you an isomorphism. And again, what you need exactly the setup, a good sign that the definition that was in the middle brief middle third was a good set of definitions is that definition is all you need to make this argument work as well. So, okay, so let me just sum up interest that that the way Bob periodicity works is the way and the way I think often to get seen, partially because mathematics is advanced so much that it's gotten kind of fractured over time and then people who learned in biology, don't talk to the people learn algebraic geometry unless they switch fields like Jim did. And, and, and it's, and so there's Bob periodicity of these maps and Bob periodicity, and they're often. They look scary, and they're often defined by analytic means because they do have clean analytic definitions not that the wrong definitions, but they're so blunt and fundamental that they come up naturally, algebraically, as well. Well, in elementary ways, by just following through sort of the exact sequences you see when talking about just bundles on people. So that is so. So that's, so that's the story I wanted to try to get across and I hope you ask questions and you can not be followed by words you just hadn't seen before because it's, it's neat how the pieces. So, thank you very much. Thanks very much for having me. I could be here and I could see other people clapping in silence.