 Thanks very much for the introduction and thanks once again for the opportunity here. In the introduction just now, Paar and I mentioned the word equivariant, and we'll see how much time we have today and whether we actually even get to that stuff. I'm hoping at least to talk a little bit about the equivariant groups. Okay, so last time we talked about the setup for the C-motivic atom spectral sequence, okay? And what you can see on the screen is the formulas, the key formulas, right? For the homology of a point, right? And for the dual steamward algebra, okay? And we talked about how this is a deformation with parameter tau, right? Deforming the classical situation, okay? And I talked at the end a little bit sort of about a philosophy that really kind of where this deformation really lifts to homotopy theories and various interesting aspects of that. And then kind of encouraging people to take another, take a deeper look at this sort of thing. One very specific problem that I hope might be solvable with these techniques is to construct the analog of TMF, of topological modular forms in the motivic context. There ought to be a K-motivic modular form spectrum over every K or some large class of fields K. And possibly this sort of deformation perspective would shed some insight into how to construct it. That's how the C-motivic modular forms spectrum has been constructed using this deformation approach and perhaps one can do this more generally. Okay, so let's take a look at some, oops, not that one, this one. Okay, I want to look at some C-motivic atoms charged to show you some of the features that appear, some of the exotic features. Okay, so what you see here, this is a C-motivic X chart. Okay, and what you see in black, the things in black are exactly the same as the classical X. Okay, and the things that you see in color are sort of exotic phenomena. Okay, the first thing you see right here is this guy H1 to the fourth and it's red because that H1 to the fourth is tau torsion. H1 to the fourth is non-zero, but tau H1 to the fourth is zero. And then there's this arrow here because there's an infinite tower of H1 multiplications and another arrow here and another arrow here and more arrows over here and so forth. And that's all H1 periodic stuff. The magenta line here has to do with the extension. It's saying that if you take this element and you multiply by H0, you don't get H1 cubed, you actually get tau H1 cubed. So there's some tau shifts in some of these multiplications and that's what those colors are doing. And if you go further, you see lots and lots of these red arrows, right? All these different red arrows that indicate tau torsion. Here are some tau torsion elements, these red dots that aren't related to H1 periodic, but they're still tau torsion. So that's sort of exotic stuff. And if you go out further, eventually, and here and now we're in the 40s, okay? We're in dimension 40 here. You see a blue dot, you see the first blue dot and that blue dot is tau squared torsion, okay? And if you go into higher and higher, and there's a green dot there, which is tau cubed torsion, and you would expect to see higher and higher tau torsion as you proceed further and further out, okay? So there's a question. I'm using the word exotic, why am I using the word exotic? What do I mean by exotic? Here, what I'm using, okay, so I use exotic loosely in lots of different contexts, but what I mean by exotic here is that it is not detected by the classical situation, okay? That's what I mean here, exotic. It's a new phenomenon that's genuinely motivic and not classical, okay? Things that are detected by TMF are things that we consider to be quote well understood, unquote, but so that's sort of exotic in a different sense, okay? So, tau torsion, okay. And one of the things about reading these charts, okay, is that this C-motivic adro-spectro sequence is trigraded. There's the topological degree, there's the motivic weight, and there's also the atoms filtration, right? And so the horizontal axis, as usual, is the topological degree. The vertical axis in this picture is the atoms filtration, and this picture does not show the weight, okay? So you have to look up on tables, you have to look up the weights of various elements if you want to know what the weights are. You can't read them off from this graphical picture. Okay, so let's get back to computing. So how would you go about making this computation that I've just shown you? Well, you can compute by machine and we have lots and lots of machine data well into the mid, between 100 and 100 to 200, around 150 or so is how much computer data we have. So plenty of computer data, okay? And you can also use a motivic version of the May spectral sequence. So I'm not going to do this in too much detail, but just to give you a sense of what happens to compare the classical and the motivic May spectral sequence. Classically, what you have is a differential that creates this relation h1 cubed plus h0 squared h2, okay? And that's what happens, okay? Motivically, you have a very, you have a similar differential, except there's a tau inserted in the formula there, okay? And that tau has to be there to make the weights balance. The weight of h1 cubed is three. The weight of h0 squared h2 is only two, and you have to have a tau in there of weight minus one in order to make things balance out, okay? So you see lots of, sort of, like you see many of the same formulas with tau's inserted, and you also see what I would call exotic formulas that don't have any classical analogs. But this is an example of a formula that has a classical analog except with tau's in there, okay? One of the interesting things about this, right? Is that then what this implies, when you multiply this formula by h1, what you get is a differential killing tau times powers of h1, okay? But the h1, the power of h1 itself was non-zero. And that's that h1 periodic, that red arrow that we saw on the previous chart. I'm just showing you symbolically how those red arrows actually, how that h1 periodic stuff actually shows up when you carry out the computation, okay? Last time, we did an extended example with the cobar complex to compute the massy product in the classical situation, h0, h1, h0, okay? And you can do the same thing in the, in the motivic situation, okay? And just to remind you, remember how these, how the correspondence goes. Classical you had the zetas, okay? And motivically you have the tau's and the x's, okay? But the tau's correspond to the zetas themselves. And the x's correspond to the zeta squareds, okay? That's what happens when you invert tau, right? Then xc becomes this tau i squared, okay? So if you make that analogy that zeta is tau and zeta squared is xc, then you do exactly the same thing, okay? I'm not gonna talk through all these details right now, okay? But if you want to, as a good exercise, right? You can go back later and I'll share these notes. You can go back later and go through and check these for yourself, right? And see how you wanna, see how this works out. But I wanna highlight, this is, this is again, with the correspondence that I just said, the tau is zeta and c is zeta squared. This is all exactly the same as what we did classically, okay? This is all exactly the same, except there's one spot where things go differently, okay? And that happens right here. You end up in the end computing that h naught comma h one comma h naught is detected by c one bar tau zero squared, okay? That's what happens when you carry this out, you get this, okay? And then this is the one place where things are different. You use the relation that tau zero squared is tau c one, okay? And you get that the massy product is detected by tau times c one bar c one. C one is h one, tau is tau, and you end up with tau h one squared, okay? So this massy product, motivically has a tau h one squared, okay? I really highly recommend going through this computation yourself and seeing at this step where the tau shows up and that's the only difference, right? And it really illuminates a lot about how things work and what you expect to see and how things can be slightly different in the motivic situation, okay? The other computation that we did yesterday was we used the May spectral sequence to compute the classical cohomology of a of one, okay? We got this thing involving the h naughts and the h ones and the v one squared and so forth, okay? And you can do the same thing motivically, okay? And again, I'm not gonna go through all the details because they're very, but they're very similar to the classical situation. But again, it would be a good exercise to go through these if you wanna spend some time after the talk, go through this and see if you can understand exactly how things work, okay? You have the same d one differential and you have essentially the same e two page except for a tau, okay? Then something slightly different occurs. We had this d two on v one squared hitting h one cubed, okay? But now it hits tau h one cubed instead, okay? So this differential does not kill off that black dot. What this differential does is it makes that black dot into a tau torsion class, okay? So it's a little different that way, okay? And then the infinity page rather than having killed h one cubed, we merely made it h one, we merely made it tau torsion. And so that's what I used an open circle and a color to indicate these black dots are tau free classes. But these red circles are tau torsion, okay? And what you get in the end is in this motivic a of one situation, you get this h one periodic stuff happening, okay? So you can see, anyway, so just a little glimmer of the first sort of exotic phenomenon of the first thing that happens here, okay? All right, so I know that I didn't go through all the details here. I just wanted to kind of sketch these for you. And if you really wanna understand these last two examples, you probably need to spend some time on your own sort of thinking, thinking things through a little more carefully. Okay, so we have talked a lot about x, there's a question. Does the motivic steam run algebra A have the usual sub hop algebras A of N, like the ordinary steam run algebra does, or their additional exotic sub hop algebras? Okay, so that's a good question about sub algebras of the motivic steam run algebra. So in the C-motivic case, I think I know the answer. You really want to, when you talk about sub algebras of the motivic steam run algebra, you really wanna think of them as M2 sub algebras, as sub algebras over the homology of a point, okay? And if you do that, then at least in the C-motivic case, you get exactly the same sub hop algebras, okay? I guess I don't know the answer for sure in the general case, in the R-motivic case, or more generally in the K-motivic case. I would expect there to be a similar classification, okay? Because everything is sort of free over the homology of a point, I would be surprised if there was anything kind of exotic happening at that level of sub hop algebras. But I'm not sure, I'm not completely sure. Okay, so the next thing we've talked a lot about how to compute X groups, all this algebra, this is step one of our three step programs. Step one is compute X, step two is compute differential, step three is to compute hidden extensions. So let's now move on to step two and talk about how to find these differentials, okay? So one thing you can do is find relations implied by total brackets, okay? And that involves the kinds of shuffling and manipulation. We did some examples of Massey products yesterday, yeah, yesterday. And you can do the same kind of thing, but with total brackets and shuffling them around and using various formulas in order to deduce relations and then imply differentials, okay? So that's one thing you can do and then that is roughly speaking what Mahowald's program was, okay? Mahowald carried out the computation of stable homogene group groups out into, well, depends how you mean, you know, let's say into the 50s, okay? And he was able to do that, like by sort of exhaustively finding these types of, he and co-authors were able to do this by finding, sort of exhaustively finding all of these, the relations implied by total brackets, okay? But eventually it gets too hard, okay? There's a question and I'll get to that in a second when I get down there, okay? So there's another approach and this approach is really kind of the innovation that has allowed us to push computations out into much higher stems in recent years, okay? We've talked about how the C-motivic category is a deformation, right? With generic fiber, the classical homotopy theory. So there's a functor here that's related to inverting tau, okay? And there's a functor here into S-mod tau modules, which is setting, like setting tau equal to zero basically, okay? And it turns out that this category of S-mod tau modules is isomorphic, or equivalent to the derived category of BP star BP co-modules, okay? So BP star BP here is sort of a package that I don't want to get into too many details because this goes outside the scope of these talks. But let me say that BP is the classical Brown Peterson spectrum, okay? That's the thing that the added Novikov spectral sequence is built out of, okay? And BP star BP are the operations, like the hop algebra of operations in that BP theory, okay? And so, but this is an entirely known algebraic thing, okay? BP star BP has been computed. It's complicated, but it's been computed, okay? It's sort of like the analogy, it plays the analogous role of the steenrod algebra in BP theory. That's the way to think about it here, okay? And you got to work with co-modules over this thing, but this is an algebraic category that can be studied by computer, right? And so the idea is that you compute in S-mod tau modules by machine because it's entirely algebraic, co-modules, okay? And then you can use naturality to pull back differentials here and then use naturality to push differentials down here, okay? And what this does is it gives you, this technique gives you a huge number of Adam's differentials up here, which typically you think of Adam's differentials as having, inherently having topological content, but we're actually doing them by machine using this comparison and that's a huge advantage. There's still differentials that we have to analyze with Tota brackets in an ad hoc way with Tota brackets, but this gives us a huge head start, okay? All right, great. And let's take a look just for sort of, you know, maybe cultural purposes or something. Let's take a look at some of these, oops, I think maybe we'll look at the classical Adam's differentials, okay? Just to give you a feeling of what this looks like. This is a classical Adam's chart because it's a little less busy and we can see a little more clearly what's going on, okay? We have similar charts in the Motiva case, okay? And what you see, these blue lines are D2 differentials. There's the first D2 differential. That's very much related. That's Adam's Hopfman variant one work, essentially that differential, okay? There are D3s showing up already here, okay? Lots of D2s in this range. These were more or less, these were worked out I think by May, okay? In the, I guess in the late 60s probably, okay? And May got stuck here, I think. He could not resolve this D3 right here, okay? Then you get into the range here. This is Mahalwal's range in the 30s and 40s and so forth. And you can see there's some D4s in green showing up here. And the things are getting quite complicated, right? Not so bad, a little bit calmer for a while, right? And then around 60, things really kind of heat up, right? And we were able to push, and you know, I don't have a chart here. This chart only goes out to 70. We have a chart that goes out to at least 90 now with all this sort of, here's a D5, a long blue D5, and you can kind of see how all this works out, okay? All right, and many, many of these, all of these differentials, these differentials we can get by machine using our deformation approach. These differentials, all of these differentials in this range all come immediately out of computer data. This is the first one, this one that, in the 30 stem, this is the first one you have to work a little bit for. That one's not that hard, but you have to work a little bit harder for that. And then it's, the next ones, I think, are really not for quite some ways, really. It's really not so bad until you get into the 50s. This blue one here, this is the first one that I think of at this point. This is the first differential that I think of as currently, genuinely hard, this first one right here. Everything up to this point now, we kind of more or less think of as easy and well known. Okay, that's a question. Does this push-pull approach, and let me pull up the slide there, this push-pull approach that I'm talking about in this diagram, help for extension problems? Absolutely. Same thing. We get a huge number of extensions using this approach that are detected algebraically, and then we have to do additional work for more obscure ones. And I'll give you some examples of that, hopefully in a few minutes. Okay, so that's kind of the story for the C-motivate case. I mean, you know, sort of talk about it in five minutes, right? But of course, it's really more like five years worth of work to kind of actually get that far and process all the data and handle elements one at a time and really kind of keep track of what's going on. But nevertheless, it's giving you the idea of how this goes. Okay, let's talk for a little bit about what happens more generally, okay? What about the K-motivic atom spectral sequence? And we're still in the situation where minus one is a square, okay? Because then rho is zero, and we have a relatively simple formula for x. We know C-motivic x in a large range, and we can get K-motivic x as long as we know the Milner-K theory of K, right? The arithmetic of K enters into it, but only kind of in this prepackaged way through the Milner-K theory, okay? So what happens, I've drawn a little schematic over here to show you what happens, how the atom's E2 page appears. The black stuff is things that we're familiar with now. This is, you know, H0, H1, H2. This is the first three stems, right? So you have the familiar C-motivic x in black, right? And then on each of these black dots, each of these black dots generates a copy of Milner-K theory, and that's what the red arrows are supposed to indicate. The red arrows indicate copies of Milner-K theory, and they could extend to the left into many degrees, depending on how complicated the Milner-K theory is, okay? But that's the degrees they line. They don't change the atom's filtration. They do change the topological degree, and they extend off to the left, okay? So that's what the K-motivic E2 page looks like, okay? And then if you look, you can see that there is room for exotic differentials, and now by exotic, what I mean is things that don't occur C-motivically, right? I'm taking the C-motivic situation as understood and saying what's new when we pass to a field with arithmetic, okay? And there can be exotic arithmetic atom's differentials, okay? And to me, this is absolutely fascinating, right? The idea that you can express this subtle arithmetic phenomena in terms of atom's differentials, I think, is an extremely powerful and probably not very well explored approach. It allows you to do things like study higher structure in these kinds of arithmetic contexts that the number theorists in the arithmetic geometers probably haven't really had the language or the tools to study. I mean, to a certain extent they have, but probably not exploited the higher structure as systematically as they could. And these atoms' differentials, I think, are a really great way of doing this, okay? So, and they really do occur, right? So Wilson and Ossphere have some examples, right? Where dr of tau and tau is sitting right there. We don't see tau because the weight is suppressed, but this black dot is a copy of m2. It's tau-free, right? It's 1 and tau and tau squared and tau cubed and so on and so forth. So tau is there kind of invisibly. And that tau can support a differential hitting something of the form A times H0 to the R where A is an element of some non-zero element of Milner-Keith theory. This really does occur in certain situations, okay? And is the argument for this Wilson-Ossphere differential arithmetic in nature? Yes, it is. What they did is they took as known information about the etalchromology of the fielding question. Prior knowledge of the etalchromology of the field and then they used that to deduce to the differential had to occur, okay? So currently, the way that the information is flowing from already known arithmetic towards the submotivic homotopy theory. I have sort of like a dream or a hope that eventually we might be able to turn that flow around, right? But that's not something that is really I think has happened yet. And so related to this next question, are there examples of these sorts of things being used by arithmetic geometers? There might be other people who have more knowledge about this than I do. I think sort of currently, I think the answer is no as far as I understand, but I'm hopeful, right? I mean that sort of thing would eventually occur, okay? Another question, does R, this R depend on K? Yes, absolutely. That's why I wrote it this way. It could be a D2, it could be a D3, it could be higher depending on properties of the field. Absolutely, okay? Their examples come from finite fields already, okay? But you would expect this sort of things to happen for a number of fields and maybe in lots of other examples as well. Okay, so I'm not going to do too much with that. The computations get much more difficult, right? Because the arithmetic enters into it. But I think that we've just scratched the surface with these kinds of computations and there is plenty of opportunity for people to do more of these types of K-motivic computations. And it's just simply ready to go. These problems, we have the technology, we have the ability, we have the background that we need. It's just a question of somebody sitting down and dedicating a significant amount of time to actually carrying out the computations and seeing how they play out, okay? I don't think we're waiting for new ideas or anything, it's just a question of somebody diving into it, okay? So that's kind of the end of the story that I want to tell for this case where minus one is a square, okay? And now I want to transition into talking about the situation where minus one is not a square, okay? And kind of like the most elementary example of that, of course, is the field R, right? And so, and this has been studied by various people, including Belmont, Duggar, Guy, and myself in various combinations, okay? So let me remind you about what the basic inputs to this computation are. Here's the homology of a point, okay? You have the tau again, and you also have a free element rho, okay? And they're connected in cool homology anyway by square one of tau is rho, okay? And where's rho showing up here? Rho is coming from Milner-K theory, right? Milner-K theory here is polynomial in rho. Rho is the class of minus one, okay? Modulo two, Milner-K theory modulo two, everything except for minus one is a square, right? And so that's sort of why rho is the only thing that's showing up there. But anyway, so that's what you start with, tau's and rho's, but there is this steam-rod operation and that's really important, okay? And then there's the dual steam-rod algebra, which takes the same form as before with the tau's and the x's, except now you have these two additional terms involving rho. We've studied a lot. These first, the tau i squared equals tau xi plus one, we talked about that relation a lot in the c-motivate case and we kind of understood the consequences of that term, right? But these extra rho terms here are an additional complication, okay? So what do you do when you see an additional complication? Well, the whole sort of like the whole theme of late 20th century and by late 20th century, I mean second half of the 20th century algebraic topology, the theme is you filter and use a spectral sequence, right? This is what you do, right? In a situation like this, when you have complications that you don't know how to handle, you filter them away and then run a spectral sequence, okay? So you filter by powers of rho and you get a roboxtine spectral sequence. That's really just a name. But anyway, you filter by the roboxtine spectral sequence and what happens is that the E1 page is really understandable. It's just c-motivate x with a free rho adjoined and that's going to converge to r-motivate x. So that's good. At least you have a good handle on the E1 page, okay? And it turns out that this spectral sequence is highly computable, okay? We can do this in a fairly large range, not as far as we can do the c-motivic stuff, but in a large range. The r-motivate x sort of turns out is readily available to us, okay? And the key idea here, I'm not going to go through very many details, okay? But the key idea in this r-motivate x in this roboxtine spectral sequence is that one thing you can do is you can invert rho a priori to begin with. If you invert rho, then what happens... Here, let's pull out this up so you can see the formula. If you invert rho, then what happens is that tau i plus one becomes expressible in terms of the other elements, okay? So you don't need tau i plus one as a generator, okay? And the result, you do need tau zero, okay? But then you don't need any of the other tau's and you do need the x's, okay? So you get f2 adjoint tau rho plus or minus one because you've inverted rho, and polynomial in tau, tau zero, c1, c2, and so forth with no relation, right? This relation gets absorbed in eliminating the tau i plus one generator, okay? And if you look at what this guy is and you study it in a little more detail, what you discover is that it's cohomology axed with rho inverted, axed in this context, axed is just the classical axed, okay? There is a degree shift here, so it's a little bit complicated how this isomorphism plays out in practice, but we know in advance, we know the rho periodic, armotific axed, okay? And once we know the rho periodic axed in advance, then everything else has to be kind of killed by a differential, okay? And because everything else has to be killed by a differential, we can do sort of a combinatorial game here and get most of the armot, the differentials are mostly forced by combinatorics, okay? Well, we already know this element can't survive, it's got to support a differential, and so there's only one possibility. Those types of arguments, very naive, very combinatorial, just look at all the data and just figure out how things can cancel out to give you this rho periodic final answer, okay? And you can do this in a long way, okay? Eventually, you have to get more sophisticated with Massey products and so forth, but not for a long time. It's highly computable, okay? After you get armotific axed, you then have to think about atoms' differentials in this context, and what we have found in practice, I don't have a philosophical reason for this, but we found in practice is that the atoms' differentials are really not that hard in this situation. Once you know the C-motific atoms' differentials, you can pretty quickly figure out most of the armotific differentials as well, using sort of naturality and various types of fairly easy arguments. There's a few cases here and there, but it's really not the most difficult thing in this whole process, okay? So what I'm saying here, right, is that the armotific situation is actually pretty good. We can do pretty effective computations in this armotific setup, okay? And this is another place where we have done a fair amount of computations, but we could go much further, and there's nothing preventing us from doing that, except somebody finding the time to sit down and simply carry out the computations. They're much more complicated, and therefore much more interesting than the C-motific case. It's really a great, great computation, okay? And again, this is ready and waiting for someone to dive in and spend it. This is not a little light side project. This is a really heavy-duty thing that takes hours and hours of work, but it would be ready to go, okay? So let me show you sort of what comes out of all of this, okay? So these charts are supposed to represent this armotific, and I've got the Adams E-infinity page here, okay? So I'm actually showing you something about homotopic groups, not about armotific X, okay? Now, I have organized these charts by co-weight, okay? Co-weight is a term that I defined on the first day. Let me write what that means is the difference between the topological degree and the weight, okay? So you take the stem, you subtract the weight, you get the co-weight, okay? Some people would call that the line, right? This is the zero line, this is the one line, and so forth, okay? We used to call that, I used to call that the Milner-Vitt degree, but I like this co-weight terminology better because it works better in the equivariate context as well, okay? So what you see here are sort of H1 multiplications, and these red arrows are infinite row towers. They go off to infinity on the left, they're row multiples, and you see the H0 tower. What you're seeing here is something analogous to the Milner-Vitt K theory, right? You see, anyway, and okay, so fine, but this is just algebra, there's no differentials happening here. Here, this is a picture of the one line, okay? So one of the things I want to point out about this picture is that you can, you know, there was some relatively recent work by, who is it? Is it Spitzfek, Rundigs, and Ostfair? Do I think about the one line? Do I have that right? I hope. Yes, you do. Okay, all right, great. I want to make sure everyone gets the credit they deserve, right? Some great work by these guys who thought about the one line over general fields, right? And sort of explaining what the one line would look like in terms of Milner-K theory and so forth, okay? And what I want to say is like, you know, if you wanted to know like how to make, how would you guess the answer in advance? I mean, proving that that's the answer, that requires hard work regardless. But the one thing you might want to do is guess what the one line looks like, or what the two line looks like, or what the three line looks like, right? And I claim that this picture here actually can allow you to kind of guess what the correct answer would be, okay? The answer that these guys got for pi one comma zero, which is really just this one column, right? Had three generators, okay? And you can see the three generators in this picture. You see row squared H2, okay? Because of that red arrow. There's a row squared H2 there. There's tau H1. And then there's this dot on the top, okay? And those three things really do correspond to the three generators in pi one zero. In the one line, okay? There's a, this row squared H2, that's got, and when you look at the formulas, you see there's a Milner K2 showing up, right? On one generator. And that's because of the, you know, it's the second, you know, it's two shifts over. It's the second Milner K theory showing up here. And there's a Milner K1, sometimes written K cross, but there's a Milner K1 that's showing up there. And then there's like a Z mod two, right? And that's from that dot right there with no Milner K3. So you could actually look at this and predict what the answer is pretty easily. And then sort of as, you know, it would be more interesting going to higher line. The one line is sort of already understood. But if you look at higher lines, right? You can make some predictions, right? You're going to see things involving new squared, right? Milner K theory times new squared is certainly going to show up, right? It has to, right? And similarly, you could make some predictions, you know, about, you know, about various things and what they should look like, right? And then you'd have to prove them. This doesn't prove this, right? But it gives you sort of a heuristic and idea of what things are going to happen, right? Okay. And now this picture looks, this one looks manageable, right? This one looks like something you could probably, you can really wrap your head around. But this one gets much more complicated, right? There are a lot of generators here and a lot of interesting relations. I don't want to dive into too much detail here. I just put to give you a flavor of how quickly things are getting really rich and complicated. The one thing I want to point out about this chart, which is really quite nasty, is this green line up here at the top of this tower, okay? Here is H3, which corresponds to the homotopy element sigma, okay? And then you multiply each, the green lines are H dot multiples or sort of like two multiples, roughly speaking, okay? So there's two sigma, four sigma, eight sigma, okay? Now, classically and semotivically, 16 sigma equals zero. That's a really important relation that has a lot to do with Adam's periodicity and V1 periodicity and is an essential sort of fact about classical homotopy, that 16 sigma is zero. But what we're seeing here is that 16 sigma is non-zero. It's detected by that guy up there at the top, okay? And what that does is it throws a huge monkey wrench into the way these things work. It's saying that V1 periodicity and Adam's periodicity are behaving in a much more complicated way than they are classically or semotivically, okay? And we really have not, this is something that, again, I think a topic that is ripe for exploration but really kind of hasn't been done. We really haven't figured out what the V1 periodic consequences of that kind of bizarre behavior are, okay? But that's a really bizarre thing, right? You know, the classical stable homotopy theorists are really like, trust me, they're really kind of, they're blown away by this, right? This is really kind of almost disturbing to them, okay? All right, so one of the things you see, I don't want to go, you know, you can go into higher and higher co-weights, right? One of the things you see is in these co-weights that are three mod four, right? You see these huge kind of lattices, right? Of eta multiplications and rho multiplications, okay? And that's a general phenomenon. Let me zoom out a little bit, right? That you can see that quite consistently that in the co-weights three mod four, the next one's 11 and 15 and so forth. And what you're seeing up here, a lot of this stuff at the top of the picture, not this noise down here, this is, but all the stuff at the top of the picture, this is supposed to be the V1 periodic homotopic homotopy, okay? Another problem, you know, as I say, a problem that's ripe for exploration, compute the V1 periodic homotopy, although there are, as I said, there are complications, okay? So this is great stuff. It's really, you know, interesting. These dashed lines are hidden extensions. We'll talk about hidden extensions in a minute, but there's one example that I want to show you just because of here. Like this is what the 12 stem, the co-weight 12, right? And you can see this really kind of complicated sequence of hidden extensions that are fitting these puzzle pieces together. There's some really, really interesting stuff happening here. Okay, we could go on for an hour just talking about this stuff, but I want to move on to other things. Okay, one of the things I want to talk about is the k-motivic atoms spectral sequence in the case now where the field does not contain a square root of minus one, okay? So what happens here is that rho is not zero. That's precisely what this formula means. This formula precisely means that rho is not zero, and so you have these powers of rho, and you have this formula for rho in the dual-steroidal algebra that you have to deal with, okay? And so again, you could try filtering by powers of rho, okay? And there are kind of two cases. One case is when rho to the n is non-zero for all n, okay? So rho is like periodic elements, invertible, okay? And then this works out relatively similarly to the r-motivic case. You're going to have r-motivic x, then you're going to have to join Milner-K theory, but I'm sure there will be exotic atoms differentials, you know, and so forth, okay? So that one's sort of, this one is, this is the case that's more similar to the r-motivic situation, okay? And then you have another case which is where rho is non-zero, but some power of rho is zero, okay? And you can still handle, you can still run a Roboxyne spectral sequence, but then it becomes more complicated because that relation rho to the n equals zero kind of creates an edge effect and causes additional complications. I decided in the interest of time to not kind of go through the details, not to discuss this in any great detail, but the point is that there are algebraic tools for getting at k-motivic x in this context, okay? And a little bit of this has been done by Wilson and Ostver for finite fields where you have something like rho squared is zero or something like that, okay? Okay, so the third part of the Atom Spectral Sequence program, right, is to handle hidden extensions, okay? So what do I mean by this, right? I've been referring to hidden extensions and showing you examples, but let me sort of slow down here for a minute and tell you exactly what I actually mean by that, okay? So the Atoms the Infinity page is not the answer we're looking for, right? It is an associated graded object of a filtration on the homotopy groups that we're looking for, okay? So what this means in practice is that when you have two dots in your Atoms chart, one above the other, okay? What that means, that means that the associated graded is Z-mod 2 plus Z-mod 2, okay? But the actual group could be, there's an extension problem there, right? The actual group could be Z-mod 2 plus Z-mod 2 or the actual group could be Z-mod 4, right? And you have to figure out which one it is, okay? So one option is to use machine data from related algebraic situations, and that tells you a lot of the answers, okay? And the other option is to fiddle with total brackets as I've written at the bottom of the screen, okay? So use the same kinds of techniques, okay? But the multiplic, but this is sort of the additive structure that's being hidden by the spectral sequence, okay? And let me just say this happens in practice, right? You get the Z-mod 4 in practice sometimes. Sometimes you do and sometimes you don't, right? It really is a question that arises, okay? You can also have multiplicative structure that is hidden by the spectral sequence. So what I mean is you might have a product, like say H1 times X is zero. You might have an element, and then H1 times it is zero in E infinity, okay? But it's possible that eta, which is what's detected by H1, and bracket X, which is my name for the homotopy element detected by X, it's possible that in homotopy, eta times bracket X is non-zero, okay? It's just detected in higher filtration, okay? And again, these sorts of things happen all the time, okay? Sometimes they do, sometimes they don't, and you have to sort them out, okay? Many of them can be gotten algebraically, but some of them, hey, you have to do in a more ad hoc manner, okay? So let me show you, I want to say one thing here. This is a little like a little fact about the classical Adams spectral sequence, a little known fact, even by people who sort of like, you'd think sort of know a lot about this. You know, we could have seen, many of you have probably seen this picture before, right? This is the classical Adams spectral sequence, right? And we're kind of, we all sort of know when the first place were algebra and topology diverge, right? We all sort of know the first place where algebra and topology diverge is right here in this first differential, all right? This Adams differential, this D2 of H4 equals H not H3 squared, it's hop from variant one differential. We all sort of know this fact, right? Everything up to that point is all detected in algebra, okay? That turns out not to be true, okay? So let me show you exactly what goes on. Let's look at this dot right here. Oh, let me just adjust this here. Let me try this maybe. Okay, let's see if this is better. Okay, what's going on here is it looks like H1's, well, what is happening, right? H1 squared H3 plus H2 cubed is zero, okay? That's the relation that you see right there, right? With the lines, right? H3 times two H1s or, you know, one times H2 times H2 times H2, right? Okay, and that makes you think, right? That A to squared sigma plus new cubed is zero. But this relation is actually false. This is not true, okay? The correct relation is that A to squared sigma plus new cubed equals A to epsilon, okay? And A to epsilon is detected by that dot there. Epsilon is detected by C0, okay? And A to epsilon is detected by that dot in higher filtration, okay? So this is the correct relation, okay? I call this, oops, Tota's relation because he's the first person that I know of, he's the first person to have discovered this fact, okay? This is the first place where topology differs from algebra, actually, okay? In the hidden extensions, right? And this is a quite complicated example of a hidden extension that happens all the way back in the nine stem, okay? So it gives you an example of the sort of things you have to be quite careful about if you really want to understand the multiplicative structure in detail, okay? All right, we are running out of time in this series and so we're not gonna do, I'm not gonna talk about much detail about these hidden extensions other than to kind of give you a sense of flavor, a sense of the sort of things that go on, okay? But I want to spend the rest of the time on is on sort of a kind of a different approach to computing stable homotopy groups that I think is also important, okay? I think that this approach using the effective spectral sequence is an important, is complementary, right? To the atom spectral sequence approach by which I mean you really want to do both approaches, carry out both approaches at the same time. There are certain features that are easily seen in the atoms context and there are other features that are more easily seen in the effective context and then when you put those two perspectives together you get a much better overall perspective on these computations, okay? So to briefly kind of set this up, right? So K-motivic homotopy theory has something called the effective filtration, okay? It's also sometimes called the slice filtration but I discourage people from using that word in the motivic context because slice filtration means something entirely different in the equivariant context, okay? The two, the motivic slice filtration and the equivariant slice filtration don't really correspond to each other. They're the same kind of thing but they don't really correspond in any concrete way and it's very dangerous to kind of be using the same word. But so I call it the effective filtration but the same thing, okay? And so because the whole category has a filtration, right? In particular, each object gets a filtration, right? And I've kind of drawn the filtration along this row, okay? And then there are the cofibers, right? The cofiber of this map is SQ, right? The cofiber of this map is SQ minus one and so forth, right? And as always, whenever you have this kind of a filtration situation you get in a spectral sequence, right? That goes from the homotopy of these slices and converges to the homotopy of X. And this is sort of like the standard setup for a spectral sequence, okay? So, but if you want this to be practical, right? You need to know how to compute the E1 term, okay? So you need to know how to compute the slices of S0, zero, okay? Well, fortunately, we do have a good description of the slices of S00. Let me emphasize this is sort of over arbitrary K, all right? And again, sort of in the interest of time I think I want to kind of skip the details of this story and I'm not really sort of an expert on this stuff anyway so there are other people who could say more. But basically you have to pass through the Motivate Cobortals and Spectrum MGL which has a very nice slice decomposition, okay? And analyze a little further. And what you get in the end is a formula for the slices of S00, okay? That involves this complicated thing, this XBP star BP, BP star, BP star, okay? Tenser HZ, okay? And what this is, this is the formal name for the Adams-Novakov E2 page, okay? So this is an entirely algebraic machine computable thing, okay? And it's the Adams-Novakov E2 page, right? So the upshot is that the slices of S00, of the sphere are expressible in terms of HZ as well as the classical Adams-Novakov E2 page, okay? And so that means it's sort of like there's, it's a practical thing to do, right? As long as you can understand something about the Motivate Comology, HZ star, over whatever field you're working with, right? As long as you can have a good grasp on HZ star, then you're in good shape for setting up the starting effective spectral sequence computations, okay? And I should say that this idea is due to Rundig-Spitzvick and Ostveer, which is, and this is a really nice result because it means that the effect of spectral sequence is practical, okay? So what do I want to say here? Let me skip that and just say this, okay? So when you look in the C-Motivate case, okay? You want to, and you should always think about that situation just as a first case to see how things fit together. What you find is that the C-Motivate Effective Spectral Sequence equals the C-Motivate Adams-Novakov Spectral Sequence, which I believe is an idea that's due to Levine, okay? But in general, it is not true that the Effective Spectral Sequence is the same as the Adams-Novakov Spectral Sequence. Obviously, those spectral sequences are related because the E1 page, the Effective E1 page has something to do with the Adams-Novakov E2 page, but they are not the same. We know from examples that they are not the same, okay? And what I want to say, and this kind of, this is sort of an important philosophical point that I want to make right here, that the Effective Spectral Sequence is really better than the C-Motivate Adams-Novakov Spectral Sequence. My feeling is that the C-Motivate Adams-Novakov Spectral Sequence is probably just too complicated and too hard to really compute with in practical terms, and that we should be using the Effective Spectral Sequence instead, okay? It's kind of like, it's sort of like the better, it's sort of an improvement on Adams-Novakov in the C-Motivate context, okay? So we saw for the sphere that, let me just pull up the formula again here at the top of the screen, we saw for the sphere that we can express the slices of S00 in terms of some Adams-Novakov E2 page, okay? And that actually happens kind of generally, okay? For reasonable choices of X, okay? The Effective E1 page is expressible in terms of some classical Adams-Novakov Spectral Sequence for some classical analog of X, okay? I don't mean this as a theorem, I just more mean this as a principle, okay? So one example of this has to do with Connective Real K Theory, Little K.O., okay? Secretly, we've actually been talking about this Little K.O. in previous lectures, although I haven't mentioned it by name, because its Adams Spectral Sequence is this A of 1, this X over A of 1 computation that we made yesterday, okay? Converging to Pi star of K.O. So we've actually talked about its Adams Spectral Sequence without calling it that, okay? The Adams-Novakov Spectral Sequence for Little K.O. looks like this. You have a, you have, the boxes represents copies of Z, and the dots represent copies of Z-Mod 2, okay? So you have the identity there, you have H1 multiples going up, you have the, you know, 2 times H1 is 0, that's the Z-Mod 2 there, and then you have a V1 squared polynomial in V1 squared as well. That's the E2 page, and then there are Adams differentials here, which of course, if you go back and you look at our May Spectral Sequence computations, you can find the analog for this differential. That's what the Adams-Novakov Spectral Sequence for K.O. looks like, okay? This is like a well-known, you know, fact from classical topology, okay? Well, the point I want to make is that if you use, you can use this well-known fact from classical topology in order to write down the effective E1 page for little K.Q., this is the sort of the appropriate cover of big K.Q., of periodic Hermitian K theory, okay? So you take this situation here, this situation where you've got the H1 towers and then also the multiples of V1 squared powers of V1 squared, okay? And you double-check what all the grating and the indices and the filtrations are, and here's the picture that you get. So in this picture, what I've got, again, the topological degree is on the horizontal axis, the slice filtration, the effective filtration is on the vertical axis, and the weights as always are suppressed, okay? So you see that here's the H1 tower, and then there's V1 squared, which is an effective filtration too, and you'll have a V1 to the fourth and a V1 to the sixth and a V1 to the eighth and so forth, okay? The red arrows indicate this, you know, that you have to have, well, here it means the homology HZ, right? Here it means HZ mod 2, right? So you have sort of, you know, things from arithmetic showing up along these red lines, okay? And then it turns out that there are, in this case for little K.Q., there are differentials, right, on tau squared, which is sort of invisible in that place, right? Up to something rho squared tau H1, and there's also a differential there and so forth. So this kind of gives you a sense, this is, as a similar feel, well, this is a good, this, you know, A of one, this K.Q., is a kind of a good test case for kind of getting familiar with the way these effective spectral sequences work, but again, I think it's totally practical that one could carry out the effective spectral sequence in a large range, at least for things like armotific homotopy theory. For sure, in armotific homotopy theory, you'd be able to do quite a bit of it, right? And perhaps for other fields as well, where the Milner-K theory is well known, okay? So there are lots of interesting things going on, going on with the effective spectral sequence that, again, these are problems that are sort of ripe for further exploration. Okay, so we're almost at the very end here, right? And it's almost a little bit, it's a bit unfair to the listeners to sort of spring a whole new subject on you at this point, okay? So let me not do any hard work here, right? And let me just sort of say some words about the C2-equivariant stable homotopy groups, okay? So we are making progress on the Adam's spectral sequence for C2-equivariant homotopy theory. As always, the inputs to that are the homology of a point and the motivic dual-steam ring algebra, sorry, not the motivic, the equivariant dual-steam ring algebra, okay? And there's a little discussion here about what the homology of a point looks like, which I don't want to get into now because we're almost out of time. But the homology of a point is understood, okay? It's got extra, it's similar to our motivic homology of a point, but has some additional stuff in it that makes it more complicated. The equivariant steam ring algebra is quite similar to the r-motivic steam ring algebra. You just have to extend scalars, right? So you start with the r-motivic steam ring algebra over the r-motivic homology of a point. You extend the scalars to the C2-equivariant homology of a point and that's your steam ring algebra, okay? And that's good news, right? What that means is that you can leverage your r-motivic computations, okay? The r-motivic computations end up telling you a lot about this C2-equivariant context, okay? Not everything, right? This extra complication in the homology of a point does matter, does show up, and that's the sort of negative cone that I'm referring to. It does create additional complications, okay? But the point is that the r-mot... Having done r-motivic computations in advance gives you a huge step up, a huge advantage to get started, okay? And these details are being kind of filled in currently. I don't wanna say anything about that. One sort of like last idea that I would like to mention, right? So this is about the atom spectral sequence and we're making progress on the atom spectral sequence in the C2-equivariant context and learning new things about equivariant stable homotopy groups. But there's another idea due to some recent work of Hanukong, right? Which is that one ought to be able to apply this effective spectral sequence technology in the equivariant context, okay? So here's what you do, right? So you have an r-motivic effective filtration of the sphere, okay? You can apply Betty realization to the r-motivic effective filtration of the sphere to get a filtration of the equivariant sphere, okay? And then you can try to run the spectral sequence associated to that filtration of the equivariant sphere. And this is like our test computations indicate to us that this is a totally practical thing to do, okay? What we're proposing here with the C2 effective spectral sequence is not the same as the well-known equivariant slice spectral sequence. It's not the same as the equivariant Adams-Novakov spectral sequence, okay? I'm claiming that it's better than either of these other spectral sequences for computing the homotobic groups of the sphere, okay? The equivariant slice spectral sequence is a great tool for computing the homotobic groups of certain types of equivariant spectra, but it's not so good for the sphere, okay? I think this effective spectral sequence is much more useful for the sphere, okay? And this is a nice illustration, right? Of how these motivic ideas, right? About the effective filtration are coming back to telling us something new, right? About a more kind of a more topological and equivariant situation, right? And sort of we, we motivate homotobic theorists ought to be kind of proud of those sorts of things when we can teach the topologists, right? Or the equivariant topologists something new that they hadn't realized before. That's sort of like a feather in our cap. That's an achievement for us, right? So a couple of questions. Okay, so is there a hope that you can do things k-motivically for a field k in order to understand things in the, I mean, I think it's gamma of k. I think my person means Galois group at equivariant setting or is the story really only for R and C, okay? So that's a good question. See the point here is that what is C2 doing here? C2 is really the Galois group of C over R, right? And so that's why our motivic has something to do with C2 at equivariant, okay? And so the question is, well, if you work over arbitrary k, could you perhaps get a Galois equivariant? So yes, it seems like a plausible story. However, the complications are going to be the profinite aspects of the Galois group are gonna have to enter the picture. And so you're gonna have to deal with all that profinite topology and all that stuff. And it seems to me like to be a very, very hairy subject. I mean, I know a lot of these things are, people are making progress on straightening out a lot of these details and that's great. But I think that there's kind of a lot of sort of underlying pro-technology kinds of things that kind of get in the way that have to be resolved in order to make a story like that actually practical, okay? Have I tried this on the omega spectrum, the spectrum omega that HHR used for the kivariant one theorem, okay? So in Hill-Hopkins-Rabinel, they kind of construct this spectrum omega using norms and sort of versions, equivariant versions of MU. And they studied the homotopy of this thing and they kind of solved this big problem about kivariant variants in classical topology, okay? And so the question is whether there's sort of saw how does that spectrum fit into this picture? So I don't know, okay? But the question that I would, the first question I would ask is whether you can even sort of like construct this spectrum omega in any kind of motivic context, right? These guys are using like equivariant tools like norms and things like that, that I don't know necessarily works so well motivically. I mean, maybe they do and maybe they don't, but that's really kind of where I would be, I would start this project. I would ask myself to what extent is the story about norms translate back to the motivic situation and see if you can kind of even define omega? It would probably be interesting to compute our motivic omega if such a thing existed, but I don't know that it exists. I have no, and it's certainly possible that it doesn't exist, right? It's certainly possible the equivariant omega is not in the image of, of, is not in the image of Betty Realization. Right. Yeah, exactly. You want to start with something involving MGL in the armative of the situation, but it's just not clear whether you can really do it in a way and then realize to, that it realizes to the ones that he'll hop into gravity. Maybe it is, I'm not saying, but I just don't have any, I don't have any evidence one way or another there. Okay. All right. So I'm about out of time and I'm about out of notes as well. So this is a place for me to stop. Thank you very much, Stan, for an absolutely fascinating lecture series. Thank you a lot. And we can have more questions from the attendees. Please ask your questions in the Q&A. Okay. There's a comment about my passion for this episode. Yeah. You know, I am, I feel very fortunate to have stumbled across this program. I mean, it's not, it's not a project. It's a program. It's sort of like a life's work at this point, right? That is like as interesting to me personally, right? I'm very fortunate to have something like that, that is just like, that's so, you know, fascinating that, you know, the stable homotopy ring is this incredibly intricate, complicated, you know, this thing that you can study calculationally. And it's just, I'm just, my personality is very well suited to this type of problem. And I'm very fortunate to have, to have something like that to work on. Could you say a word about how the effective spectral sequence comes about? I'm gonna, I think I'm gonna decline this question. It's sort of, there's sort of a long story here. And I'm not super familiar enough with the details off of top of my head to really be able to say anything intelligent. There are certainly people in this video conference right now who would be able to kind of free, you know, to speak freely about that sort of thing. But like, but I'm not the person to do that. Sometimes it's best to know when you're not, you know, to know when to retreat, right? And this is one of those situations. Could you say one more time how the C2 effective spectral sequence comes about? Yeah, sure, sure. So what you do, right, is you have, you have the, let me just write it here. Right, so you look, so you work in the r-motivic context, okay? And so what you have, right, is the, you have an r-motivic filtration here, right? SQ of the sphere, SQ minus one of the sphere, and so forth, okay? So you take this whole diagram, okay, and then you betty realize it, okay? And, you know, and you get something equivariant, right? So these guys, let's just call them, you know, they don't necessarily have a name. They don't necessarily have much of a meaning except that they exist, okay? But what happens, and then the same thing with the, with, with this little column capital S for slice or something, right? Same thing with these things, okay? Except the thing is that, you see, we know that things like hz go to equivariant hz. And hf2, motivic hf2 goes to hf2. And the motivic sphere goes to the equivariant sphere, okay? So the point is that you actually know what the, see here, here, we know these, these motivic slices in terms of, of Eilenberg-McLean objects, right? They split as Eilenberg-McLean objects. And we have a good handle on, on what those things are. And so therefore, these layers are also built out of Eilenberg, equivariant Eilenberg-McLean objects, okay? So we can actually, even though we don't have good categorical names for what any of these, for what any of these objects are, we do have a perfectly fine filtration of the equivariant sphere, whose layers are in terms of equivariant Eilenberg-McLean objects. And so we can write down the E1 page and compute. And of course, this map, this, this Betty Functor is going to, by, you know, it's going to induce, by naturality will induce maps between differentials, right, in the, in the effective spectral sequences. So differentials over here will, will imply differentials over here and, and vice versa, to a certain extent, okay? Mark, does that answer your question? Yes, very good. Yeah, thanks very much. Okay, yeah. It's a really, I think it's a really kind of, It's not a really, it's not a sophisticated idea, right? But it's, but it turns out that it's, you know, it's, it's, it's, it looks to be surprisingly useful. So let me follow up on that. Is there a version of the deformation theory description in the real Betty case? I would love to know the answer to that question. There is not currently, but I think there ought to be, and that would be, and that would be a great, that would be a great thing to have. Okay, sounds like an interesting question. Yeah. Okay. Do you think it is possible in the future to know all of the homotopy groups? Well, it sort of depends on, okay. So it sort of depends on what you mean by no. So first of all, there is this thing in stable homotopy theory called the Mahowald uncertainty principle. Okay. And the Mahowald uncertainty principle says something like, I forget there's, there's a kind of a nice way of phrasing it, but this essentially what it says is this, is that any algebraic tool, right, or any kind of finite collection of algebraic tools that you have for approximating the stable homotopy groups is going to kind of eventually fail, right? And have, and then you'll have to do some like, find additional techniques, right? So there's no sort of purely algebraic way to describe all of the, there, someone in the chat, someone has quoted the text of it. Thank you. And so from that, and I certainly agree with that philosophy. However, notice what it says. If you look at the text of it, it says something specific about using, that can be named using homological algebra, right? So maybe there are other more sophisticated algebraic tools that we don't know about, right? That have never been thought of or discovered that could potentially be used to describe all of the stable homotopy groups. I would not rule that out. I think that there's, you know, the potential for that kind of thing, but of course it's highly speculative, right? And we just don't know. Okay. Is there a motivic story of genera, like there is classically, which allows us to interpret motivic stable stems as target for comportism invariance of schemes? I don't know. You know, I mean, the role of MGL is sort of, is more complicated than the role of MU in topology, right? So it strikes me as a sort of thing that's likely to have some serious difficulties, but I don't really know. There are non-cellular objects in motivic homotopy theory. What is their role in this picture? Could they be approached by an atom's spectral sequence? So the role of the non-cellular objects in my picture is that essentially they don't exist. Okay. I've been talking a lot about motivic homotopy theory, but really I've been working in cellular motivic homotopy theory. Okay. I'm looking at the subcategory of objects that are built out of spheres. Okay. And this is good and it's bad. It's good because it makes the category much more tractable to computation. It's bad because it doesn't capture all of the geometry and arithmetic that you might want to study, right? Things of invariance of elliptic curves are stuff that motivic homotopy theorists ought to be caring about, right? Computing these sort of, you know, homotopical invariance of an elliptic curve is definitely something we should care about. And that's basically inaccessible from my perspective. Okay. The problem is that these types of spectral sequences that I use aren't going to converge in those cases. Okay. Or they're going to converge to a cellular approximation of, you know, of something like an elliptic curve, which is going to miss a lot of the interesting geometry. Okay. So that's sort of like the limitations of this method for sure, like getting at those types of things. Can you provide an idea of what sort of motivic information is lost when you pass through the Betty Realization Functors to the equivariant setting? Okay. So the real information that's lost when you pass from R-motivic to Betty Realization is exactly what I was just saying. All that sort of geometric non-cellular stuff is for sure going to be lost when you pass to the C2-equivariant setting. Okay. But in terms of the cellular information in the R-motivic homotopy and C2-equivariant homotopy, I think that there's sort of, is a reasonably good way of summarizing that. And there's a slogan here that I didn't have time to talk about, but it's written here right in the middle of the slide. Right. So there are results of Barron's and Shaw that make precise this idea that C2-equivariant homotopy theory is the tau periodicization of R-motivic homotopy theory. Okay. And what that means is a little bit complicated because tau is not actually even an element, is not a map in R-motivic homotopy theory. It doesn't survive the Roboctyne spectral sequence, but it's sort of like tau is some sort of like, you know, some periodicity operator. Okay. Or some bracket or something like that. And so they make precise this idea, right, that C2-equivariant is the tau periodicization of R-motivic. So what you lose, right, is, you know, the tau period is the tau torsion. Right. Whatever tau torsion means in this context, that's the sort of thing you lose. Okay. So one specific example of that is the eta periodic homotopy. The R-motivic homotopy category has a rich eta periodic theory. Okay. Where you can compute the eta, we've computed the eta periodic homotopy groups completely in the R-motivic context. And you get a really interesting complicated example. Equivariantly, the eta periodic groups essentially collapse. There's like, there's the identity element and so it's non-zero, but just barely. Right. Basically, there's the identity element and that's it. Okay. So the eta periodic story equivariantly is much less interesting. Okay. And what this slogan is saying is that all of this eta periodic R-motivic homotopy theory is tau torsion. Okay. And then so it's not being detected equivariately. That's sort of roughly speaking. But again, remember also, all of the non-cellular stuff will also somehow disappear when you go past the equivariates. Okay. Is there any application of motivic methods to complications of symplectic co-boardism ring of the point that you know? Not that I know of, but yeah, sure. Compute the symplectic co-boardism ring of the point. Okay. I think that concludes all the great questions. So let's thank Dan once again.