 Yeah, so I'd like to thank the organizers for giving me the opportunity to speak here and for organizing conference. So yeah, I'm going to be talking about vibration theorems for fatigue and pollution of structuring spectra. What I'm hoping to do though is draw a lot of analogies to a much more classical story which takes place just in spaces like good old fashioned spaces and good old fashioned homology. So this does come with a disclaimer like Duncan's talk. I'm more of a homotopy theorist, but I'm hoping to keep things a little bit grounded. So first we're going to just talk about this classical notion of completion for spaces. And the idea here is that homotopy groups are often very hard to compute, but homology is often easier. So you might have the idea given a space X, I want to build the part of that space that homology sees. And I'll say more specifically what I mean, but let's assume we could build such a gadget and let's just call it X bar. There's a couple of desirable properties. If X bar is supposed to be capturing just the homological information, we would want that construction to not be able to distinguish between two spaces with the same homology. So if I've got a map X to Y, inducing a homology isomorphism, I would want this construction, I would want them to be weakly equivalent. And I should point out from now on, NEH star UC really means reduced homology with integer coefficients. If we're trying to build a part of a space that homology sees, though, we want some sort of comparison from the original space to this new construction. So we'd want a map from X to X bar and at the very least, we'd ask that it be a homology isomorphism. So how could we build such a thing? I'll just remind you this is what we're sort of after. And a good place to start, excuse me, is with the Heritage map. That relates homotopy to homology. And one way to construct the Heritage map is as follows, I can start with the following assumption. So when I say space, I really mean pointed simplicial set. If you're not familiar with that, it's just a delta op shape diagram and sets. And it's a model for regular spaces, there's equivalent equivalence. So how this adjunction works, well, each set I can take and form the free abelian group. So that would then give me a delta op shape diagram and abelian groups, which is also what we call a simplicial abelian group. So that's the left adjoint here on top. And then, of course, I could forget the extra structure of being the group. I could forget back down to being a set and that's this right adjoint U. And like any adjunction, this comes with a unit map. So I get a natural transformation from the identity functor here to this UZ tilde gadget. And now I started this whole discussion talking about the Heritage map. So where does that come in? Well, to this map, I could apply homotopy, by pi star, and ask what happens. And what happens is I get the Heritage map. So there's this natural isomorphism between the homotopy, the UZ tilde y, and the homology of y. And this composite is exactly the Heritage map. What that tells us is that this construction, UZ tilde, satisfies that first property that we wanted. That's not too hard to check. On the other hand, it's also not too hard to pick up an example that shows we usually don't have the second property. We usually do not have a nice comparison from X to UZ tilde X. But it's a place to start. And to improve this, to build the completion, I'm going to just sort of fatten this construction up. So I'm going to iterate this construction. So remember, we start with this adjunction. I have this unit map. This is sort of this Spaces level Heritage map. But what we can notice now is that I could stick another unit map here or here, like there's two ways to do it. So I would get, that's better. We would get two maps up to UZ tilde, UZ tilde. And then again, I could stick another unit map here or here or here. So I'd get three maps up to this UZ tilde cubed. And I haven't teched it because it gets a little gross. But the co-unit map associated with the adjunction gives me a map down. And I actually get two maps down here. So this whole thing is a delta-shaped diagram. And that's good because we have lots of tools to deal with those. And so to define the completion, I want to just sort of glue together all of this stuff in some homotopy meaningful way. And that's just really what homotopy limits are for. So I take this red thing and I glue it all together. That's just if you're not familiar with homotopy limits, just think of this as sort of homotopy meaningfully gluing stuff together. And we call that the Z completion of a space. And this eta map induces a natural comparison from X to its completion. So this is our stab at constructing the part of the space that homology sees. And these are the things we wanted. So I wanted these properties. And one can check that, yes, I have the first one. That's good. The second one is more subtle. But surprisingly enough, often you get something even better than just a homology equivalence. Often you get a weak equivalence. And this should be a little surprising because remember, this completion is just coming from iterates of these guys, which only see the homological information of X. So somehow, if I glue together in the right way things that only see the homology, I can actually recover the space up to weak equivalence. That's just up to homology. And that leads us to ask the related question, well, when exactly does this happen? When can I rebuild a space X out of much, much simpler components by just gluing them together in this formal process? And Bousfield and Kahn in the 70s gave an answer, which is that this happens if and only if a space X is nilpotent. The key step in their proof is understanding what completion does to certain vibration sequences. So in more detail, a space is nilpotent if and only if morally it can be built out of a sequence of principal vibrations. They show that completion preserves these principal vibrations. So they ask, well, when does a fiber sequence remain a fiber sequence after completion? And that it's an inductive argument up a post-Nikot counter. But the real technical heart of what they do is understanding how completion plays with fiber sequences. So now I want to introduce that's the classical story. And now I want to retell this story in a newer context of structured ring spectra. So I'm going to talk about completion of O-algebras with respect to P2 homologies. This is very much in the same vein of Duncan's talk. But the thing to keep in mind is this is an analogous story of Z-completion in structured ring spectra. If you're not familiar with spectra, I hope hopefully Duncan's talk enlightened that some. But you can just think of spectra as sort of a better version of spaces. Like the theory of spectra encompasses the theory of spaces, but they can have much richer algebraic structure on them. So remember, for spaces, we started with this adjunction, some partial sets, the central of even groups. And this is the analogous adjunction that we're going to have in structured ring spectra. So algae O, what this morally is, is just spectra with extra algebraic structure encoded by a reduced operand. So this is exactly the category that Duncan was talking about. So we can think of commutative ring spectra, associative ring spectra, EN, all different kinds of algebraic flavors. Because this is so general and so algebraically rich, though, it's not necessarily the best category to be working in. So what we do is go to a nicer category, which is this algae tau 1 of O, because tau 1 of O throws away a lot of information. So it agrees with O at level 0, but other than that, it's trivial. So we've lost a lot of information, but the upside is it's a much nicer place to work. So again, this is the adjunction. This is the analogy we've been thinking of. And the TQ topological and homology of an O algebra, we define as what I get when I just take somebody in here, go over to the nicer category, and come back, just like we define ordinary homology of a space. In the same way that we tried to build the part of a space that homology sees, like ordinary integer homology, we can build the part of an O algebra that this TQ homology theory sees. And we do it in the same formal way. That is, I take my O algebra A. I have my unit map that's acting like a Haravich map. And then I iterate it to form this cosine-pushal gadget. And then to define the completion, I just glue all that stuff together. So I glue together these TQ homology spectra. Just like in spaces, we can ask, well, when can I recover, excuse me, my O algebra A up to weak equivalence by doing this? When can I throw away a lot of information, but still through some formal process, recover the space that I started with, or in this case, recover the O algebra that I started with. Ching and Harper give in 2019 sufficient conditions. So they say if A is zero connected, then I can recover this. But that's not a complete characterization. They don't give a complete characterization. And we expect that we're gonna need to know, just like basketball and con, what does TQ completion do to fiber sequences? So the motivating questions of this project is, well, okay, how does completion play with fiber sequences? Or even as a warm-up, are there O algebras that are not zero connected? So not covered by Ching, Harper, but which are nevertheless TQ complete? And it's worth noting basketball and con's proof really relies on a characterization that's unique to spaces, that I have a free group action on the space, and that's something we don't have in our setting. So there is definitely a new proof required. So we're still working on a complete answer, but this is the theorem that I wanna talk about, which is that if you hand me a fiber sequence in O algebras, where the total and base objects are zero connected, then after I hit it with completion, I still have a fiber sequence. And so with the remaining time, I just wanna give you a flavor of the proof, we're gonna skipping details left and right, but just wanna give you sort of the strategy and attack here. So the first thing we do is consider sort of the canonical diagram. Well, I started with this fiber sequence FED, and then I look at what happens when I complete those. We know these maps are weak equivalences because E and D are zero connected, that's the Ching Harper result. What we don't know is this left-hand map. I don't know if that's a weak equivalence. And to show this theorem, to put this theorem on, it suffices to show that this is a weak equivalence because then this bottom, right, the fiber sequence after completion is weakly equivalent to the fiber sequence we started with. So that's now the strategy. I wanna show in this case that F is complete. So how are we gonna do this? Well, this is the thing we're interested in analyzing. So let's try, let's remember how I built this. So how did I build the conclusion of B? Well, I took B and I iterated that hereditary map to get this cosimplitial gadget, same thing with E and same thing with F. So now you say to yourself, well, okay, this left-hand column, that's a fiber sequence, life's good, that's what I gave myself. The issue that you run into right away, though, is that this next column is not gonna be a fiber sequence. There's a couple ways to think about that. Either you can say to yourself, well, I started with a fiber sequence, which is a limit, and I hit it with this Q-funker, that's a left adjoint, generally left adjoints don't play well with limits. Another way to see it is, well, I took a fiber sequence and I applied homology, and in general, that's really non-trivial to ask what happens. The serospectral sequence gives us some idea, but then you're gonna have this infinite sequence of serospectral sequences, not to mention the fact that we don't have the serospectral sequence in our setting. So this diagram is not quite what we want because we lose control of it too quickly. So what we're gonna do instead is just take homotopy fibers vertically. So whatever the homotopy fiber of this map is here, I'm gonna stick that here, and same thing here, and same thing here. What this gives us then is this diagram. The good news is that each column is a fiber sequence, so I have a good amount of control over the diagram. However, that came at the cost of having the thing we were interested in being the diagram in the first place, right? The thing that constructed the completion is gone. So somehow I need to get the completion of F back into the picture, and to do that, I'm gonna just consider this top row here, and stick it at the bottom of something. Now I was interested in getting the completion of F in here, so let's just draw it. There's the completion of F. And then the natural thing following your nose is to resolve each of these other objects with respect to TQ. And then I have all these maps here. Now we're ultimately interested in analyzing this map, right? I wanna say that that's the equivalence after applying Ho-lim, and I'm gonna conclude something about that by looking at some other maps. So that is to say, first I look at these vertical maps. It's fairly formal that these give you a weak equivalence after applying homotopy limits. I then look at these horizontal maps, and through some careful cubicle diagram analysis, I'm able to conclude that these induce a weak equivalence on homotopy limits. And it's sort of a two out of three argument to then say that this map that we were really interested in is also a weak equivalence. So what we get is that in this case, F is TQ complete, and that proves the theorem. I should also say what we get is actually a little bit stronger, which is that it's not just a weak equivalence after applying Ho-lim. I can say something about the various stages, which is that I get a proisomorphism here. And I also, which should finish by mentioning that there's a generalization of this that we're able to provide, which is that TQ completion preserves pullback squares, not just fiber sequences of this form if the X and Y are zero connected. To get the fiber theorem result, all you need to do is replace B with a point. So this is sort of a first step in the program of completely characterizing TQ complete O algebras, and that's definitely the subject of future work. And that's all I have. Thank you very much.