 Okay, so let me first of all say that I really very much appreciate the effort that the organizers and also that IHES have gone have put into converting this event into a virtual format. We all know that this is not the the style of event that we were hoping for that we were that we were wishing for but I think it's really important that all of us at all levels of our society under these circumstances do the best with this with in the circumstances that we have even if those those efforts are not ideal. Okay, so let's go ahead and get to work. So I'm going to talk over the next three hours about stable homotopy groups. Okay, so let me begin with some background. Okay, mostly classical right about the stable homotopic groups and why these are things that we should care about computing. Okay, so what so so s not right this is the name of the unit object in the stable homotopic category. Okay, and it's it's the unit object in the sense that if you smash any any spectrum x with s not you just get x back again. Okay, so it's the it's the unit of that object and the stable homotopic groups are basically by definition they are the graded endomorphisms of s not they're the graded endomorphisms of the unit object that's what we're talking about right and as we know from many examples the endomorphisms with the unit object control the structure of the entire category right so these pi star actions these actions by the endomorphisms have a lot to do with the structure of the entire category. Okay, so for one very concrete example we could think about the the class of two cell complexes so two cell complexes are the of the spectrum that you get by taking a sphere a sphere spectrum mapping it to another sphere spectrum and then taking the cofiber okay and then the the and then this sk right that's the shift of this this this one to begin with right okay so so these these two cell complexes x of course they're in correspondence with the elements of pi star right there's this map that you took the cofiber up to get x right so if you want to um classify two cell complexes you have to compute the stable homotopic groups okay so that's one sort of like very naive way in which um these sort of pi star uh the these elements of pi star tell you about the whole about other parts of the category okay and so you might want to take this example of two cell complexes and extend it further right so what about um it's so in general and in general finite cell complexes are very much related to the structure of pi star okay but in a more complicated way okay in a more sophisticated way so let's take a look at as for an example of that let's take a look at a three cell complex so i've drawn a picture over here on the left which is sort of a schematic of what i want to do right so i want to have um a i want to build a complex right and it should have s mod beta it should have the two cell complex associated to beta as a sub complex at the bottom okay and it should have the two cell complex associated to alpha as a quotient okay at the top right that's what this picture means sort of schematically okay um and so you can do this um sometimes sometimes you can build a three cell complex um where the bottom two cells are s mod beta and the top two cells are s mod alpha however it turns out there's an obstruction you need a condition okay and that condition is that you need the product alpha times beta or the composition alpha times beta as endomorphisms of s mod of endomorphism unit object that composition needs to be zero okay that turns out to be an obstruction to constructing such a three cell complex okay and then you could go further right from this three cell complex and you could ask about well what about a four cell complex so a four cell complex so the schematic looks is is like this picture i've got over here on the left okay and again the idea is that we're building a four cell complex it should have a certain three cell complex as uh three as a sub complex and a certain other three cell complex as a quotient right and so and those quotient alpha beta and gamma okay so we already know from the previous example that you have to have that alpha beta is zero in um in order for that top three cell complex to exist and we also know that beta times gamma has to be zero for that bottom c three cell complex to exist okay and then it turns out there's an additional obstruction there's another obstruction to actually putting all of these things together into a single four cell complex and that has to do with the total bracket alpha comma beta comma gamma okay you need this total bracket to vanish or at least to contain zero okay so i haven't told you what a total bracket is okay and we will get later in these talks we will get into a little bit of this sort of these sort of higher operations this higher structure that uh that one needs to uh to study here okay but i but we won't be probably too precise about that the point i want to emphasize here is that there's higher structure that you need to know about to solve sort of real tangible problems okay and then you can try to take this sort of cell complex idea to an extreme right and maybe now i've looked at some what you know uh uh you know sort of a very much more complicated type of cell diagram whatever this even means right um and you could ask whether you could form and a cell complex of this type over here on the right and then it turns out you know um there are some obstructions right what are the obstructions well um you know to constructing this thing um and that turns out that there are but they involve something that maybe you would call mixed length brackets to get even more complicated okay so on the one hand this higher structure like for example in the four cell complex situation on the one hand this four cell complex this this higher structure does a very nice job of understanding how four cell complexes exist but also things kind of get spiral out of control fairly rapidly when you try to study the general situation and this is more or less equivalent to the fact that the stable homotopy groups are complicated and the stable homotopy category is complicated and we wouldn't expect there to be necessarily a simple classification of arbitrary finite cell complexes okay so here's the conclusion that i'd like to draw from from these examples and from this sort of discussion first of all the most yeah the important thing that pi star is not just a graded abelian group it is a graded abelian group okay but it's much more than that okay it's also a graded commutative ring because you can compose endomorphisms okay but it's much more than a graded commutative ring as well okay the higher structure is an indispensable indispensable part of the structure of pi star okay you haven't really understood pi star unless you've understood all of this higher structure okay and that's something that people who you know spent time making explicit computations of stable homotopy groups spend a lot of time worrying about this higher structure digging into it because it reveals so much of what's of what's going on okay and we'll talk about that higher structure at various points along the way okay so everything i've said so far was sort of background motivation about classical stable homotopy theory okay and let me point out that uh that much of this same story applies just as well in motivic or equivariant or other contexts okay the motivic stable homotopy groups homotopy groups or the equivariant stable homotopy groups will control finite cell complex constructions in those contexts as well okay however there's an important um there's an important caveat here especially in the motivic context right which is that not every motivic object is built out of cells okay is cellular in the sense of built out of spheres and so you so these stable homotopy groups are good for the cellular objects but they're not necessarily so good for other types of object objects okay the good news is that many of the most important motivic objects like the algebra k theory spectra the island bird mcclean spectra or the um the the co-borders and spectra and so forth are cellular okay so it's still so just so studying the cellular objects is motivically is still a worthwhile thing okay all right so let's talk a little bit about sort of like the background about the contexts in which we're going to be working okay so in the upper left corner I've got a little diagram here right of four categories and four functors okay in the upper left corner to cat corner I have the r-motivic stable homotopy category okay and then in the lower left corner I have the c-motivic stable homotopy category and those two categories of course are connected by the extension of scalars functor right okay and then in the upper right corner I have the c2-equivariant stable homotopy category and in this situation here c2 is really the Galois group of c over r that's sort of why it's c2 as opposed to some other group okay and Betty realization maps goes from r-motivic homotopy to c2-equivariant homotopy okay every r-motivic spectrum has sort of sort of sort of an underlying c2-equivariant spectrum and that c2-action is the Galois action right okay and then finally in the lower right corner I have the classical stable homotopy theory okay and stable homotopy category and then that receives functors the forgetful functor from c2-equivariant just forget the c2 actions okay and then of course the Betty realization from c-motivic homotopy theory okay so these four categories fit together very nicely and these we should think of these functors as being sort of very well behaved computationally we can really kind of understand them if we set our minds to it okay so the program that I you know I'm proposing or I'm working on is that we should be working we should be computing in all four of these contexts simultaneously okay because the because the way they relate along these functors tells us a lot of information that the situation becomes much more rigid and much more easy to understand if we actually do all of these at once okay we're also going to consider k-motivic stable homotopy groups for some sort of general class of fields k all right but maybe in less detail than the r-motivic and the c-motivic cases okay so let me defend that that choice here that uh let me defend that choice for a minute of why focused on the r-motivic and c-motivic cases rather than the uh than the general field okay so uh the uh so uh the the point for me is that the r-motivic and the c-motivic cases are the ones that are most closely related to classical homotopy theory to classical topology okay so if you want to learn something about classical topology or if you want to borrow tools from classical topology then the r-motivic and the c-motivic cases are the places where you're most likely to be to be effective okay so there's that okay but also maybe even more importantly the r-motivic and the c-motivic cases are more accessible and so they're more they're good they're important tests for general theory okay and a great historical example of that is what um is what happened over the last few years with eta periodic homotopy okay so eta motivically the element the half map eta is not nilpotent and we'll talk more about that in computational detail later okay but it's not nilpotent which means you could invert it and still have something non-zero okay so you invert uh you invert your eta okay and you see what you get you compute what you get okay and there was a series of of of projects um uh with burke you and myself andres and miller uh glenn wilson kyle ornstein oliver rundig's tom bachman mike hopkins in where we started at the beginning of this series we started looking at the c-motivic computations and we figured out what happened there and then we went to the r-motivic computations and we figured out what happened there and that led to um ideas about what the general picture should look like over the rationals and then in over general fields okay so there sort of was this progression from specific cases that gave us hints about the general theory to the to to the general case okay and that's exactly the way it played out um and so that's just that that that's that's sort of an important principle here and why we want to work through i'm all i'm saying here is like we should do the easy cases first right that's that that's somehow summarizes this whole this whole point okay so um let me make some comments about sort of standing assumptions and uh in in general sort of philosophy for uh for the series okay so first of all i'm always working stably i may not always say stable homotopy or stable homotopy group or stable homotopy category but i always mean stably everything we're going to do here is stable okay lots of stuff are um going on lots of stuff going on here unstable in motivic homotopy theory but that's not the subject of these talks okay question are there other motivic fracture squares um analogous to the real and complex version so i think what okay so i think what's being asked there is that back to this back to this upper left square are there analogous squares like this for other fields and the answer i i if i understand the question correctly i would say the answer really is no right there's something very special about the real numbers and the complex numbers and the way they relate to ordinary topology one could try to do things with the at all homotopy types and so forth over general fields um and that would probably you know and then general galois groups and that would probably be a fairly interesting thing to do but of course there's all that complicated technology pro technology that has to go into that sort of thing and i don't know how like say computational that would necessarily be okay but but that's but working out you know those ideas is probably and a lot of those things have been worked out right it's probably you know it's probably a worthwhile thing okay and the other question is for the c2 equivariance is this naive equivariant no i mean the so-called genuine equivariant theory i'm thinking about um representation spheres and stabilizing with respect to all representation spheres okay all right okay so we're always working stably usually we'll have completed in a prime okay and usually that prime will be two okay there are some places some parts where things work integrally but if you're going to do computations as a general rule you have to work one prime at a time that's just the the price you pay for actually getting compute explicit computations out of things okay um and you know and and there are ways of reassembling all this you know this primary data into you know integral stuff with the you know with all the usual sort of like technology okay um the other standing is another standing assumption that i'm going to make is that i'm studying the stable homotopy groups here there's another perspective on what sort of like these the fundamental invariance of the of the motivic homotopy category is and that uses the idea of studying homotopy sheaves okay and the and the this homotopy sheaves are more powerful and that they can then help you study objects that are not cellular and so you can do much more interesting geometry and arithmetic with them the the downside of course is you you lose a certain amount of explicit computational ability when you're working with these abstract sheaves okay um and the relate the connection between them is that the homotopy groups that i'm studying are the global sections of the homotopy sheaves okay so that's what we're going to be talking about okay um and i sort of as i already alluded to before we will complete as necessary to make whatever spectral sequence we're studying will complete as necessary to make that thing converge that might mean completing at a prime p that might mean completing at that hop also completing at the half map eta you might have to do something where you take the effective completion of of a spectrum and so on and so forth okay so i am not going to make a big deal about this out of this completion stuff okay uh generally speaking it it um it there is some work to be done about these convergence issues and about the behavior of these completions and and typically this work is manageable okay uh it's it's not trivial but it's manageable right and so we can get these spectral sequences to converge in in in reasonably nice ways okay my job is not to worry about the convergence my job is to sort of figure out what the computations are right and so that's what we'll talk about okay question um is it then known that uh about what about a homotopy sheaf version of the motivic atoms spectral sequence so that is a good question off the top of my head i have never thought that through maybe some other people here have some idea that my instinct tells me that it would that it should work just fine uh the problem is that in abstractly you should be able to set up such a thing just fine that my however it's not at all clear that you're going to be able to make this sort of fundamental computations to get things off of the off of the ground okay um and then sort of related question there um what what you lose specifically when you work with the homotopy sheaves is that you that maybe the spectral sequences exist but there's there's sort of there's another thing that you need besides the existence of the spectral sequence the other thing you need is some input computations that you have to start with right and so what for example when you're thinking about the atom spectral sequence and we'll get to this in a little bit but so this is a little bit of a preview but when you think about the atom spectral sequence you can set it up abstractly but that's only useful if you know what the comb algebra point is and you already know what the steering algebra is if you have no idea what the steering algebra is right then then the atom spectral sequence is nothing more than an abstract toy okay so that's the problem with with with the sheaves right if you're going to sort of work with the sheaves you probably going to I don't I I don't know I don't want to speculate right here live about what you're going to need but but I have a feeling that those sort of those input computations are are just kind of like you know not really things you can write down okay all right um so uh we'll complete as necessary and then finally one last comment here about um of just about notation is that the grading convention that I will adopt is you know in this form p comma q this is the this is the grading convention that babowski used p is like the topological degree q is the motific weight and then p minus q is some is frequently a quantity that one wants to study and uh and I'll call that the co weight okay because it's sort of a partner to uh sort of a partner sort of dual in a sense to weight okay and this does not agree uh with the notation that all um authors have used on the in this subject but it's the one that I'll stick with okay consistently all right so even before we get we are certainly headed for the atom spectral sequence that is the sort of like the first big tool that we're going to use but even before we get to the atom spectral sequence let's do let's go back to sort of like prehistory even before that before the atom spectral sequence was used to study stable homotubic groups there were some more geometric constructions okay that that work in sort of very low degrees okay so let's talk about that style of um uh of constructing stable homotubic elements okay so and some of these ideas are due to many of these ideas are due to morrell some are written down by dugger and myself um and then who in creche as well have contributed contributed at various points along this way okay so these geometric constructions the good thing about these these constructions is that they are universal okay they work over spec z and therefore automatically are going to work in in sort of over any base okay in the in the motific context over any base okay all right so the first element that i want to discuss is the element row okay in pi minus one minus one okay so this element row can be constructed you take plus or minus one and you complete it concluded include it into gm okay and just as a matter of notation gm here i just mean a one minus zero i take the affine line i puncture it and then that's gm that's that's you know for the multiplicative group whatever but we won't really need that g well anyway it's gm it's just it's a more convenient notation you include plus or minus one into gm you get something that turns in some cases turns out to be non-trivial okay and we'll call that row okay more generally you can include one and some unit u into gm okay and get an element of of pi minus one minus one that we could maybe call bracket u okay and row is another name for bracket minus one okay row comes up so frequently that we give it its own name and these bracket us in general are a little more obscure okay so these are already some geometric constructions okay closely related to the arithmetic of the field right okay um oh and why is it minus one minus one right because this is an s zero zero and this is an s one one and then the relative degree is minus one minus one okay um then there's an element epsilon in pi zero zero so that's the twist map you have gm smash gm you have a symmetry right swap the factors to gm smash gm the relative degree there is zero common zero that's in pi zero zero okay and because it's a twist map not surprisingly this epsilon controls commutativity and we'll write down a formula in a minute for what exactly i mean by that but epsilon is is essential if you're going to want to study some form of commutativity okay so now so the those are sort of like the elementary like most naive you know some of the most naive things you could think of okay and now things get a little more interesting okay and so you borrow an idea from very classical topology right from um from at least as far back as hopf right okay so um and you can construct a hopf map eta okay in pi one comma one okay i'm going to construct eta in a way that's it's probably not the most common it's probably not the way that most people who have seen this before think of of of eta okay but it's it's it's useful for a certain perspective okay so eight is in pi one one so here's what i do okay start with gm cross gm okay and it has a multiplication map to gm right gm's a group right and then suspend it once that's what the suspensions are okay so this mu is really suspension of you okay so there's that map mu okay it turns out for very general reasons okay after one suspension a product always splits this is a very general fact about about homotopy theory and and so this so this this product splits and one of the summands of this splitting is the suspension of gm smash gm okay so there's this inclusion here right that comes from this very general categorical splitting okay so now you have a map from suspension gm smash gm into suspension gm okay that composition and you go and you count the degrees what do you have here you have one two three spheres two twists right so three comma two here you have two spheres and one twist and so you have two comma one okay and then the relative degree is is one comma one okay so there is there is a map eta okay and that map that's the same the way that people usually think about eta as the projection from a two minus zero down to p1 okay and this is the same map or maybe it's off by a minus sign but it's essentially the same map as um as that construction okay uh and oh the other thing you know I should have said this at the beginning actually you know all of these geometric constructions that I'm doing here are are are unstable right I'm actually doing unstable constructions here and then stabilizing them right in order to get stable homoteles but this map eta really exists in s3 2 to s2 on unstable okay and so so there's eta okay and then you know from classical topology that the half maps don't end at eta that there are higher dimensional analogs of these things and so we'll take a look at the next one okay new all right so here's what you can do with new you can take the group sl2 okay uh that's of course a group and then and you have the multiplication map right after one suspension from s suspension sl2 cross sl2 to suspension sl2 okay and you mu all right and again this categorical splitting right gives you a map from suspension sl2 smash sl2 into suspension sl2 okay now here's the interesting fact it turns out that sl2 has the homotopy type of s3 comma 2 okay that's not a hard that's a relatively easy geometric thing you can do you just look at columns that have to be you know uh and determine it has to be one and you can kind of contract you know things down and get that equivalence that's not a that's not a very hard fact but it's an observation okay and so then sl2 is an s3 2 sl2 is an s3 2 and you go up and you count degrees and you end up with a map from s7 4 to s4 2 okay and then that gives you a construction new that works unstably and it works over any over any possible base okay finally something weird happened here hang on a second this is supposed to be s7 4 okay now with sigma a new complication arises a new the new complication is that you can't model sigma as this kind of hop construction on a group object okay what you have to use is a non-associated multiplication okay but you can show that over any base s7 4 comma 4 has a non-associated multiplication okay and then you use that non-associated multiplication in the same way as before using the that splitting to get a map from s15 comma a to s8 comma 4 and that's pi 7 4 okay uh so great so that's the classical stuff right and now we know from classical history that you can't really expect to go much further at this sort of naive level right that um that there's something maybe more sophisticated if you really want to go further you're something you have more sophisticated that you have to do okay but before we dive into those more sophisticated techniques let's talk a little bit so we've got these elements rho bracket u epsilon eta new sigma and let's talk a little bit about relations amongst these things okay so who in creche proved a very nice result um it's basically a steinberg relation that u bracket u times bracket one minus u always equals zero okay and epsilon squared is one that epsilon was the twist and so if you twist twice of course you get the identity okay here's the formula i wrote it down here for the formula for graded commutativity if you want to compare alpha beta and beta alpha what you have to do is possibly put in a minus sign right and possibly put in an epsilon factor okay depending on the degree of alpha and beta okay the exact formula here is not so important right now you can look that up later if it's a formula that you want to use but the point is if you do the kind of like the diagram chasing and you figure out exactly what happens when you swap factors around you have a plus or minus one and maybe an epsilon um in there as well in order to switch things okay and that's sort of a really interesting wrinkle uh classically we see the minus one we don't see the epsilon that's a really interesting uh wrinkle okay um you can show that row times one minus epsilon is zero you can do this geometrically you can draw you can construct complexes and show that this factors through something contractible right same thing with eta times one minus epsilon um eta times new and new times six you can you can show all of all of these relations here you can show them geometrically by constructing you know uh complexes that constructing objects that are that are contractible that these compositions factor through okay and of course um for those who know about this stuff what you're seeing on this slide is a lot of information about Milner-Vitt k theory right uh some of what's going on in Milner-Vitt k theory is appearing in some of these formulas Milner-Vitt k theory saying even more than that and I don't want to get into that um uh in in these talks but I just want to sort of make a nod to that whole circle of ideas that you can um that that that you can develop them them further okay so instead I'd like to go in a different direction okay so um how might you go deeper how might you produce more stable homotopy elements okay well one thing you could do is you could follow Tota's classical work so Tota carried out some sort of amazing stable homotopy group computations with like with really very little technology like without using things like the atom spectral sequence he was able to go remarkably far into the structure of the classical stable homotopy groups and you could try to follow the approach that he the kind of approaches that he adopted for example you could use Tota brackets okay so a Tota bracket is a way of building new stable homotopy elements out of old ones it's kind of like composition but more sophisticated okay and we've talked a little bit about that earlier this is part of the higher structure of the stable homotopy groups okay so one the first example that um that that that occurs is this Tota bracket that I've written down on the screen so eta comma one minus epsilon comma new squared okay it turns out that we know geometrically we know that eta times one minus epsilon is zero you could I already wrote that relation down and then it also turns out that one minus epsilon times new squared is also known to be zero okay and that those two relations make this Tota bracket defined in pi eight five okay and that thing exists that's again this is all over spec z right this is in the universal or even unstably right you can even make this unstable total bracket this all works in complete generality okay and you could try to go further right but it gets harder and harder and more and more ad hoc and it's just not sort of you know um well you could do it I just you know and but this is as far as people have I think really gone in this direction and again I again I do think that people could go further if they decided to sit down and and think it through okay so this is kind of the end okay all right so now we come to kind of talk to a turning point in the history of stable homotopy groups right with the advent of the atoms spectral sequence okay so the add of spectral sequences of course due to atoms right but it was also one should give a certain amount of credit to ser as well right so ser had these had these some ideas about computing state computing homotopy groups by this method of you know of using island room co-mology of island room reclaimed spaces and to a large extent what atoms is doing is systematizing and organizing the kind of ad hoc approach that um that um uh that that ser was sort of trying to kind of like trying to describe okay question do we have motivic moho old root invariance defined so yes there are um root invariance running around in this story there are a few different ways that you the a few different things you could mean by that okay and the uh the uh the the short answer is that you should look at uh JD Quigley's work JD Quigley has written a couple of papers I think about motivic homotopy theory and root invariance and he has shown how to construct these things in some level of generality that I forget off the top of my head and he uh has carried out some computations right and kind of indicating what maybe these things are good for okay uh and there's another sort of let me just since we're on the subject of root invariance let me also say that um one of the sort of ongoing projects that that I and some of my co-authors have is to use our motivic homotopy theory to further our knowledge of classical root invariance like I think that our motivic the homotopy theory can beat the um can beat the classical topologists at their own game that we can do better at at computing um root invariance if we use a little bit of motivic homotopy theory okay all right um and it looks like in the chat that there was a link posted to uh to JD Quigley if you want to know more about a motivic root invariance that's a that's a good question okay so we're going to talk about the atom spectral sequence okay this is um supposed to be a summer school right and this is the first week of this summer school and so I decided that um to spend a a certain amount of time kind of covering um covering you know what's really what's really background right and so I want to talk in some detail about how about what the atom spectral sequence is about how you construct it and why this particular construction ought to be something useful and interesting okay so this this next part of the of the talk is really all classical review okay and then I'll say some things about the motivic and equivariant um variations you know that that come up maybe at the end okay so I'm going to write h for hfp right remember McLean spectrum and a prime okay and I write p but really in my own head I think p equals two because I'm always working at at two but but I guess we don't need to do that here okay and h star right the coefficients of h star is um is fp right that's easy enough okay and then the other thing we need is h star h or in other words the homotopy of h smash h okay and that's the dual steamer in algebra a star okay which is a hop algebra okay so it has a multiplication and a co multiplication okay so this hop out this this a star it's kind of complicated I'm not writing down the formulas for it right now although we will write down formulas later okay but the point is that it isn't completely explicit it is completely known okay the other thing that I'm doing is that I am always writing the dual steamer in algebra I am never talking in this entire series I am never going to talk about the steamer in algebra I'm only going to talk about the dual steamer in algebra because it turns out that that the computations work out much more nicely in the dual case of course in some philosophical sense they're equivalent all you're doing is dualizing over a field but uh the the formulas are much nicer to write down uh in in the dual situation and so that's one of the one of the real what one of the early obstacles that a lot of students have to diving into this subject is making that transition from the steamer in algebra which is more natural psychologically to the dual steel steamer in algebra which is much easier to work with in practice okay so um that's something that you kind of have to train yourself to spend some time doing and you have to train yourself to think in those dual terms okay so there is a unit map from these uh from the sphere to H okay and it has um and then that gives you if you take the fiber you get a co-fiber sequence and that's the definition of H bar okay so H bar is like the difference between the sphere and H okay and that looks like sort of just sort of an arbitrary thing there there's no motivation for that okay but here's a little bit of motivation right if you look at the homotopy groups of H smash H bar right what you end up doing is you end up taking eight you get a bar right which is the augmentation ideal of the dual steamer in algebra so H um H smash H bar has a nice algebraic interpretation right it's a topological thing right but algebraically it's corresponding to taking the augmentation ideal okay all right so here's so that's the ingredients that we need okay so here's how you construct an atom's resolution okay so you start with here's this co-fiber sequence that we just talked about on the left those two maps make a co-fiber sequence okay and then if you take that co-fiber sequence and you smash it with H bar right take each of these three objects and smash them with H bar you get these three objects okay and so those three objects also form a co-fiber sequence okay and then if you take those three objects and you smash them again with H bar you get those three objects and you get another co-fiber sequence okay so in this picture each of these L shaped these these three terms in a shape of an L form a co-fiber sequence okay the the the row itself is not any kind of exact thing it's it's it's more like of a resolution or something like that okay all right so whenever you have this kind of sequence of nested co-fiber sequences right you end up with a spectral sequence okay the spectral sequence starts with the homotopy of these third terms this H this H smash H bar H smash H bar smash H bar and so on and so forth starts with the homotopy of these third terms and it converges to the homotopy of S0 okay another way of thinking about this is that here's S0 and you filter S0 along this tower this is like a filtration of S0 and then these are the associated graded these are the layers of the filtration like the associated grade that's another good way of thinking about okay and that's what a spectral sequence does right it goes from the layer it passes from the layers to the whole object right and so that's exactly what you get here so the E1 page of this spectral sequence has the homotopy of all these guys right and it converges to the homotopy of the sphere okay and then to make this converge you need some p completions here right for convergence and that's okay you've chosen a p up at the very beginning here right and so um so there's some convergence there but that's that which again you know as I've said is manageable right there's some things to do but it's manageable okay so this looks fine right but what's really going on here why would you do this why what what makes this sort of anything useful other than just some like arbitrary like you know crazy arrows that I've written down on the screen well it turns out that this E1 page is totally computable right we know a lot about h smash h bar h smash h bar I wrote that down earlier that's the augmentation ideal okay and it turns out when you smash with more powers of h bar it it still is computable okay and so what you get in this E1 page this E1 page I've written it out here more explicitly okay you get an f2 that's from h that's the homotopy of h you get an a bar that's the homotopy of h smash h bar and then you get the second tensor power of a bar that's what this homotopy turns out to be that's not very hard it's a little bit of a computation and you can get that that's the second tensor power of a bar and then the third tensor power and the fourth tensor power and so on and so forth okay so this E1 page this has a name this is this is called the cobar complex of a okay we'll talk a little we'll talk in more detail about this thing later and carry out some computations okay what this thing is is a differential graded algebra whose homology is is the x groups of the ring a over with coefficients and f2 comma f2 okay so this cobar complex is sort of like is it's kind of it's a fundamental object right it's um it's it's it's a key tool for computing x okay and this observation I think is really now that I've written this down I think now you can go back and you can look at the motivation for what the atoms resolution is doing okay when you want to study the higher invariance of a ring f2 because you're taking the prime two yeah exactly that's a typo that could be these can be those those two should be p's at this level I just I always forget because I literally like I eat and breathe and sleep p equals two and so I just I'll I constantly forget that okay um so uh so when you want to study the higher invariance of a ring we know what to do we take a resolution and we take derived you know x and all that and tour and all that sort of stuff right we take we we do that sort of thing and the cobar complex is a nice convenient tool for those resolutions and doing those kinds of derived constructions okay so what's happening here in this atoms resolution is you're doing what you're trying you're playing out that same story right of looking um of taking resolutions and looking for higher invariance but instead of doing it in algebra you're doing it in topology you're using the specter themselves to build the resolution right but you're really mimicking the algebraic situation here in topology okay so that's a good kind of one a good way of sort of motivating of wrapping your head around what the atom spectral sequence is really trying to do okay then sorry to have interrupted you there is a question on the chart uh one more p on the e1 page um I don't see it right here here oh yes you're right okay thank you that should be a p as well great okay all right so um the upshot here is that the e2 page turns out to be um is x over over a fp comma fp okay and then that's converging to the stable homotopy groups okay that's the kind of like that that's kind of like you know the the consequence of having done all of this okay and that's kind of like the key that that's I mean somehow some sense like that's the thing that you need to remember from all this if like you didn't wrap your head fully around what all of this um um if you didn't fully wrap your head around this whole construction of the atom spectral sequence and where it comes from and what it's motivated by you don't necessarily to worry too much about that what's important is that there is a spectral sequence it starts from x groups and it converges to the stable homotopy groups okay and we're not really going to dive into any of the details of the construction in the rest of these of these talks but we are frequently going to be talking about x groups and how they're related to stable homotopy groups okay so this one this this this formula right here at the bottom is really kind of like the thing that we need to carry forward with us okay so here is the program the program is first compute those x groups okay that's an algebraic exercise we know a explicitly we know fp we we can do that x groups explicitly that's that's algebraic then it's a spectral sequence and this spectral sequence can have differentials and it does have differentials okay so you have to analyze the differentials in the atom spectral sequence okay and then finally you get this e infinity page but then there's some interpretation of the final answer and that is involved the solving extension problems okay we're going to talk in great detail about each of these three parts of the program but this is this is how it goes there's always these three steps you need the algebraic input you need to analyze the differentials and then you need to interpret you need to analyze the hidden extensions interpreting the final answer okay so everything I've said over the last few minutes was entirely in the classical context okay but these all work just fine came to this is a pretty general setup here right and it works just fine cameotifically or gequivariantly um and uh the key point is you need to know about the cosmology of a point or the homology of a point I guess and you need to know the dual steward algebra explicitly if you know the dual steward algebra explicitly and you know the homology of a point explicitly then you're ready to go you can start an atom spectral sequence project okay uh and and uh there are um there are comp there are additional complications with convergence in this motivic or equivariant context but these are the the uh they are manageable they're requires it requires real work but these these things work out okay and so various people who have worked on these construction these sort of foundational stuff for the motivic or equivariant atom spectral sequence include morrell dugger myself who increase who increase in ormsby um maybe maybe some others as well okay question is this x in modules or co modules is there a difference okay um so what I um this is a good question and this is an important point and I always get sort of tripped up about this and then there's tor and cotor also and all the duality okay so what I'm thinking of here is x in modules okay notice when I wrote x sub a here I didn't write a star I wrote a okay and so that's what I mean I mean a is a ring fp is a module over a and I'm taking the derived functors of hawn in the category of a modules okay that's what I'm referring to here okay um you know when you take x right what you do is you take a resolution for f2 right for fp in the first variable right and then you haunt it into fp right and then you take homology okay and the cobar complex is what what you get the cobar complex is the thing that you when you take a free resolution of fp and then you haunt it into fp what you get is the cobar complex okay and then the homology of the cobar complex is x over a okay so that so so that sort of duality is kind of built in when I take the cobar complex it's that haunt into fp that's happening there okay so that's the right way to think about this you can set this all up in the category of of co modules okay the category of co modules is like dual is equivalent to the category of modules in some sense then you can set it all up that way and change the names of things um but but let me just leave it there that's saying I mean x of of of a modules x to modules okay uh but all right so what we need to learn about right is the co homology of the classical steward algebra in other words that's the name that we give to x of a um f2 f2 so now I'm writing p equals 2 here because I want to talk in detail about how the computations play out okay so that's the first thing we need to do we need to dive in to this algebra and and and study with study this okay so in the 21st century the way that we study these x groups is by machine okay um computers love to do linear algebra and we can ask the computer to construct minimal free resolutions of f2 as long as and it will it will hum along for a few months and produce all kinds of great data some people who um are closely associated with this idea are uh bruner nassau and guo xin wang who at various points have have written and implemented effective software for doing this okay these computations are effectively implemented in a very large range okay out to like you know at you know say maybe 200 or more stems okay far beyond our ability to interpret it okay and that will always be the case right we'll um so we should take the computer data essentially as as given right we have as much computer data as we want okay so I am going to switch now let's so let's take a look at what this ends up being okay and we'll talk more about where this comes from um uh in in later points but for now I just want to dive in I want to sort of look let's just look at some data okay so what you're looking at here is a classical x chart okay or an add or you know a classical atoms chart okay um what you you do see off to the right you see some blue and red lines those are atoms differentials which I don't want to talk about now we'll come back later and we'll look at this chart again it's I this was the chart I had available and so I just used it right but we want to kind of ignore those those blue and red lines and just just look at the black dots and lines okay so this is what you get right there's this huge bi graded group okay and it starts off looking in low dimensions it starts off looking not too bad okay um it's you know it seems sort of manageable right and even in this range right there aren't too many dots it sort of seems manageable right and as you go out further things get more complicated but still not too bad it's a little getting me a little bit crazy around here um and then things get worse you know and you sort of you know maybe more I mean regularly if I zoom out a little bit I can show it this chart goes out to 70 we have charts that go much further than that but that that obviously but going out to 70 kind of like proves the point when you get out into this range things get and again we kind of want to ignore the colored lines but even just look at the number of dots like right here in this degree there are three different dots right so things get a little bit complicated okay you can see some regular patterns right like if you look up along the top you see this a regular repeating pattern there okay and you also kind of see some parallelograms that kind of regularly repeat along here and there's some there is some regularity at the top of the chart and there's a lot of noise along along the bottom okay so this is what happens you start with the steam run algebra you start with f2 you compute x and you get this thing right that has structure that has um you know periodicity that has some regular structure but also has a lot of irregular structure okay and that's what you expect as you go into higher and higher stems you expect to see more and more complications more and more irregularities end up occurring okay all right so that's not meant to be so there's no detail here right of course and and that's not the point this is sort of more like a cultural kind of presentation rather than anything um you know in uh in in specific but to give a sense of what's going on um I should mention well while we're looking at this and the things we've talked about before um this guy here h0 that's the that guy detects the element 2 okay and that's 4 and that's 8 and so forth h1 detects eta in pi 1 h2 detects nu in pi 3 and there's sigma in pi 7 okay we talked about eta nu and sigma the motivic versions of them but these are the classical versions eta nu and sigma in pi pi 1 pi 3 pi sigma pi 7 okay and we also talked about this guy this bracket in pi 8 5 right and we talked about this sort of like way you could construct another element in pi 8 5 and that's corresponding to this element right there called c0 okay that's how I knew to write down that particular total bracket because I knew that c0 that's the next thing that you might be interested in right and so you could so and so I wrote down a bracket for that next thing and then ph1 you could try to write down a bracket for ph1 and ph2 and d0 and so forth right and these would be perfectly worthwhile things to have have construction stuff okay so you're already kind of picking up a lot of of information just by looking at this chart kind of qualitatively without even worrying so much about where things um exactly come from okay so now the other thing I want to do and again this is ah okay sorry um the question about the degrees here right so the the vertical axis here is the atoms filtration okay um and the horizontal axis axis is the stem the vertical axis is the atoms filtration and that's how all of my charts will be organized okay so this is a classical chart so there is no weights okay it's just c0 in pi 8 here okay um turns out it's it's it's weight is 8 5 okay um when I do show you motific atoms charts later they will not they'll the weight will be suppressed right that that the the motific atom spectral sequence is tri-graded there's the topological degree there's the weight and there's the atom filtration well I I I can't plot it in three dimensions I've tried and it doesn't work so I have to suppress the weight okay and so you would so pi 8 5 will appear right here and you won't see the 5 you'll have to kind of you have to go into the computations or look up the tables and see what the weights of these these these elements all the weights are known and they're in tables but they're not displayed on the charts okay that's a great question about about about how the charts are laid out and where the weights are okay so now what I want to do is I want to show you a different thing I want to show you this okay so I want this this maybe I'm just sort of showing off here by uh okay you guys should be able to see a window okay so this is a chrome browser and it is displaying an app that uh hood chatham uh wrote so or is writing so this thing is a um this the hood is writing a spectral sequence sort of analysis tool okay it doesn't do the hardcore computations the hardcore computations are done elsewhere and then import it into this interactive tool okay so I can scroll around on this thing right and I can click on a bi degree right and then I click on that bi degree and over here it lists on the right side it shows me the name gives me the names of the classes and it tells me something about the products okay I can move I can move elements around if I want to for example if I don't like the way these things are located I can go in here and I can I can switch their locations like it just did okay um and and and and and so forth okay so this is still in relatively primitive um state it's not ready for public release but it is making progress and I thought I would like to show it off and put in a plug for the for the great work that that hood is doing this kind of a tool is going is is a great way of sort of keeping track of what's going on you know when you get into higher stems right and there ended up being so many elements right after a while that you really need a good way of keeping track of things right and all these different relations right and this is this nice interactive tool for really studying things we intend eventually to allow input atoms differentials and so forth and make it a really nice scratch pad for carrying out special sequence computation so that's something that's I'm hoping coming in the next year or so a product that you know that that that I certainly want to use and that maybe other people who are carrying out these kinds of explicit computations would be interested in as well okay so let me go back now to the presentation okay okay um so what I want to do next is talk about how you you know studying these x computations and more detail what's really going on in these x computations okay and a good that sort of like the most naive way to tackle this these x groups to begin with is to study the co bar complex okay so um I want to say a little bit of we're almost out of time for today so I'll say a little bit about this now and then we'll pick this up again I think on Thursday the next talk but I will start talking about it now okay so first of all what I I've been talking about the dual steering algebra but I haven't told you what it is explicitly okay so what it is is a polynomial algebra over on generators zeta 1 zeta 2 zeta 3 and so forth okay and this is a computation this is due to Milner okay um but this thing is a hopper algebra not just an algebra okay it's got a product and a coproduct okay and the coproduct I've written down a formula for the coproduct over here on the right okay so this is a complicated formula with a lot of moving parts and not so easy to understand and I don't expect you to sort of stare at this thing and memorize and wrap your head around it fully well one thing to remember is that the coproduct in the dual corresponds to the product in the steamward algebra so in the steamward algebra you have these adem relations right that have to do with the product structure the adem relations are somehow encoded in this coproduct information here okay the product the nice polynomial product over here corresponds to the coproduct in the steamward algebra that's the carton formula the carton formula is a nice regular thing in the steamward algebra and that's corresponding to the nice regular polynomial structure here okay one thing that I want to point out is that um the what are the primitive elements the primitive elements are elements whose coproduct is just themselves tensor one plus one tensor themselves okay and the elements that are primitive are precisely zeta one two to the end okay um and and and that and that's it okay if you take anything else other than zeta one to a power of two uh then you're going to get something that's not primitive okay so that's kind of a an important point that we'll see in a minute okay so the cobar complex is we've talked about this f2a I guess I mean a bar okay and then um a bar tensor a bar and so on and so forth okay and if you dive a little deeper into the cobar complex and you look at what these maps are you discover that this first map is the coproduct okay so if you want to actually compute the homology of the copar complex which is exactly what we want we want the homology you need to understand this this um this coproduct which I've written down here okay and we'll dive into this um next time okay so next time we'll we'll go back a little bit we'll we'll set up the copar complex again and we'll dive more into computing um x groups and we'll carry out lots of explicit examples okay I think that's a good place to stop for today okay many thanks indeed and let's thanks the speaker and so are there any questions please raise your hands over the chart please there is a question yeah are there people working with motivic um he wrote an ss which is adams nova cop spectral sequence for other prime spree well and adams nova cop spectral sequence is something to work on but what we've been talking about today is adams spectral sequence at um at other primes so um not so much okay there are some sort of there are kind of there are some philosophical reasons to anticipate that that while the motivic adams spectral sequence has been sort of really interesting at um at the prime two that the at odd prime somehow things are a little bit more kind of just like the classical story with some extra weights thrown in and a little bit of curiosity okay there are some indications that that has been or let let me let me rephrase that let me say the conventional wisdom over the last 10 years has been that somehow the odd primary computations will be like classical with a little kind of like you know perturbation a little a little you know a few little wrinkles um and that the p equals two computations are somehow much more um uh much more fundamentally interesting i over time i'm becoming less and less convinced of that conventional wisdom so i consider it's sort of like this odd primary computations to be a sort of a wide open subject i think there's a lot of room for someone to dive in and and really sort of tear these things apart and see what's going on and have a good and try to get a better understanding of of what's happening there so um there is some work you know um uh but there's really nothing like real well developed okay other question any special homology theory such that sigma or nu is not zero okay um so i question um i i think the idea here is that um can you so so the thing about you know um the ordinary homology is that it detects one right and sort of nothing else right and um and maybe and but then something like ko complex uh sorry not complex real k theory detects the element eta right and then the idea is that could you go further along those lines and find something further more complicated than ko that detects nu and detect sigma okay so it depends on what you're looking for but one thing that detects nu um nu is is tmf okay and that's sort of one big reason why tmf is so interesting is that it captures eta and nu and the various higher consequences of having eta and nu at hand okay and and yet it throws out all of the additional complications that occur with sigma and in in higher places like so for example when we wrote down that form that total bracket for that guy in pi eight that what was it eta comma one minus epsilon comma nu squared well that's a thing that's built out of eta and nu and two and things like that and so that guy does is detected by tmf okay because it's it's associated to those eta and nu type family right um and so that's one answer is that that's kind of like from my perspective that's why tmf is so interesting other people have other reasons for it and that's and those are important reasons also similar work regarding the equivarian case yes absolutely we are in the midst of cranking through the atom spectral sequence um at for the c2eco variant atom spectral sequence anyway and we are making progress i'm i'm hoping by the end of the third talk to at least talk a little bit about that that's time permitting we may or may not get to that but absolutely one can one can see um there's work there by um brit g u and myself um michael dug ravinell brit g u and myself um and hana kong has also made some progress along those lines um millner vit k3 know about nu and sigma um new millner vit k theory does not know about nu and sigma i wrote on that slide here i can share that let me find that slide i wrote on this slide i'll see also millner vit k theory what i meant by that was that many of the formulas that i wrote down many of the constructions and formulas that i wrote down in this section are related to millner get millner vit k theory not all of it in particular the nu and the sigma are not in millner vit k3 but the eta the row the one minus epsilon that bracket you that part of it is really sitting inside of millner vit k theory i'm sorry for that that that was my fault for writing a confusing slide okay um equivariant dual steam ring algebra for p and odd prime so yes um lots is known about the equivariant um steam ring algebra uh at at even primes and at odd primes the thing to remember is that you have to have a group as well okay so it depends on what group if you're working with c2 then i think we know both the odd and the you know we we know the the two primary steam ring algebra we also know the odd primary steam primary steam ring algebra but the odd primary steam steam steam ring algebra is not very interesting it's c2 c c2 is a group of four or two we expect the p equals two computations to be more interesting than the odd primary computations for cp for an odd prime p for cp i think that this is now known um and maybe in slightly more generality what i would look at is um i would go i don't have the top of my head i don't have the answer but i would go search for um things that that igore creche and various co-authors have have written have been writing recently have been writing about this and i forget exactly what but there is sort of an ongoing program to expand these kind of steam rod algebra co-mology of point computations into larger and larger classes of of groups okay um and unfortunately things get kind of really complicated really quickly and it's and we're not really yet we're not ready yet to dive into atoms computations with those types of things yet because they're they're just even c2 is really giving us a real uh giving us a real challenge and the bigger groups are just going to be a nightmare at this point although eventually we'll get there but but it's it's it's that's off in the future um atoms types picture sequence based on chow wit co-mology i have not thought about that um i'm not exactly sure what you mean by chow wit co-mology i think maybe you mean something like kq um or kq with a to inverted or something like that um so maybe you're kind of getting at some sort of version of you know this at so this bo resolutions um yeah so yeah i think you mean kq with a to inverted then maybe maybe um i i don't know um what that reminds me uh uh uh the question shan tilson's question reminds me about about chow wit co-mology reminds me of is this this the story of bo resolutions so mohoalden co-authors um attempted to mimic the atom spectral sequence but instead of using co-mology h ordinary co-mology they used um they used something like ko real k theory and they tried to carry out an atom spectral sequence type analysis for ko okay and they got a fair ways right and they saw some interesting structure that you couldn't see otherwise uh so it was partly successful but the computations also get very complicated uh and so it was as it was only partly successful some of that ko story is now being developed in the motivic context so who is this so dominant culver and jd quickly and maybe some other people as well um are are are working on that sort of of of thing okay mark levine is saying that is chow wit is like a version of c2 bradon co-mology yeah then i i'm not really sure how chow wit fits into uh fits into this it's not something i've thought about okay um the co-mology of a point how does one go around these computations okay so um oh sorry i'm jumping here wait uh let me let me go in order can you explain what e1 and e2 definition are here is the spectral sequence defined in different way to co-mology theory okay so i think the question here with the is with the atom spectral sequence right how is e1 how are e1 and e2 defined okay so e1 is defined simply by taking the homotopy of the homotopy groups of these third terms right of these layers okay that's all it is it's just that that's the definition of the e1 term okay and it turns out that we know what the homotopy of these guys are and this has and and it's expressible in terms of the co-bar complex okay so there's two steps here formally e1 is this and then computationally we know that this equals something in terms of a bar about this about the augmentation ideal of the dual stereo algebra okay um then uh then the the e2 page is defined to be the homology of the e1 page there is a differential on the e1 page that goes from um from one term to the next so here's how it goes if you start with something like this right remember this is a co-fiber sequence okay so there is a map there's a shift map right the boundary map from this guy it goes maps here let me draw it in color there is a a shift map that goes back there right it goes through the suspension of that thing okay and then so you can you can apply that and then apply that map and you get a map from here to here and the same thing from here you have to apply the the boundary map and then down and so on and so forth and that map that's the d1 okay so you when you take the homology of e1 with respect to that differential you get the e2 group the e2 um and that turns out to be x and again that's a computation right so formally it's given by this composition but it's a computation that it works out to be x over a okay and then um about the motivical mongeva point how does one go about those computations okay so the computing the motivical mongeva point is a very deep very difficult problem right this is the blockado conjecture right so vavadsky did these computations okay um that is i do not feel qualified to talk about the details of that uh sort of sort of thing uh here uh and so i'm not going to say anything about it and it's certainly not something we're going to get into in any detail however let me foreshadow something that i'm going to say more about tomorrow that in certain cases now we have a kind of a way of working around all of the deep difficult mathematics that vavadsky did to compute the motivics theorem algebra and to compute the motivical mongeva point okay so in certain cases in particular in c motivical mongeva theory um for the kinds of things that i do we don't really need vavadsky's computations anymore we have another way of accessing the same results okay using deformations of homotopy theories and i will specify a little bit more about that in on thursday's talk in my second talk on thursday okay let's thank the speaker again and so well uh the next talk is in 90 minutes maybe less by tina gerhardt at six o'clock peris time okay um i'm seeing a comment i i'm not in a hurry to stick around to to leave and this channel stays open right yes okay um i'm seeing a comment from from paul arna that shan means millner vitt motivical homology so right so that makes a lot more sense to me um i don't know that's an intriguing idea i don't know i mean i know of this mo millner vitt motivic homology i don't know much about it right tom bachman who was here who may or may not be here right now has been here sometimes this week probably can tell us a lot more about that right um so that's an interesting question uh my first question is what about the co-operations right what about um do we know anything about the operations in millner vitt motivic homology because uh again for explicit comp that that may or may not be the most important question for people who study millner vitt motivic homology but for the purposes of explicit computation that would be kind of you know an important question right and i guess maybe the idea would be then the atmospheric sequence you know in this context then completely captures millner k theory right um something something something like that might might be the way that that works out i don't know good question sounds like an interesting project but i don't know enough to what to i don't know i don't know enough to assess whether it's realistic or not i'm not saying it isn't i just i just don't know uh then another question from indy baker yeah i see that yeah all right um i'm getting a question sorry discussions time now yeah yeah yeah i mean it's it's fine this is fine with you if i keep going right as long as there are questions yeah in fact i'm uh i'm here during the day okay great okay great um uh um so i think that the difference between new bar and totus epsilon right is atus is atus sigma okay and so atus sigma is a multiple of eta and so it's in the indeterminacy so the actual answer andy is that it detects both of those and um there's yeah there's sort of well i could go on i'm gonna i'm gonna restrain myself right because i could start kind of like i could i could riff on this subject for a long time right but but um but it's both because of that and you have to straighten one of the tricky things you have to do in in the eight and nine stem is you have to sort out the difference between them and there's actually kind of some hidden structure there that you need to kind of be really careful about the difference between new bar and and and epsilon um and yeah depending on how you look at it you kind of see it in different ways is there a way that we can see the question yes shan i think if you go in the q and a and you click on answered there's like a tab for open and a tab for answered and the answered ones are there is there a way to detect the non-triviality of a given framed manifold um right so this question about framed manifolds goes back to like the early early history of stable homotopy groups right there's this this close connection between framed man co borders and some framed manifolds and um and stable homotopy groups right and there was some work i mean i think maybe even like pi three right classical pi three maybe it was even studied this way right before it got to be sort of too too impractical um so i think that in that range you can be you know up to like pi three say i think you can be explicit about how stable homotopy elements interact with framed manifolds and my understanding is that once you get beyond that that things really break down that you really can't you know people don't really know uh how to sort of write things down in those terms um in a motivic or equivariant context i'm not even quite sure how that's going to play out i mean i guess there's a lot of recent work right about how framings relate to motivic stable homotopy right and that seems like kind of a promising direction but um you know again because my sort of specialty my interest is in explicit computations i don't know to what extent that's um uh um uh i don't know to what extent that's kind of like you can you can do anything explicit although i don't know maybe go back and look at the old you know work of you know i know rocklin and i forget um there's another name associated with that that i'm drawing a blank on right now but uh go back and look at that old work and see whether there um uh whether you can kind of come up with the algebraic versions of the kind of like the manifold constructions that those guys were those guys were studying okay good all right there's a reference for um millner-vitt-based um adentetrical sequence that's yeah that's promising uh equivariant motivic homotopy theory i mean so you know inevitably one is going to need to take equivariant motivic homotopy theory seriously right there is some work i'm thinking of people like ormsby heller who increase right and maybe others who have been sort of laying out foundations and some you know um preliminary steps in the direction of equivariant motivic homotopy theory for sure there are interesting things there um i'd probably be doing myself if i didn't have like a lifetimes worth of back backlog of other problems that i want to solve first um i i i think that's i think it's a great direction to go in um i think the key idea about equivariant in the key idea is sort of like to choose the right kind of goal right what are you trying to do with equivariant motivic homotopy theory um and one of the things i would like to understand to know more about i think that we should know more about is the is the quadratic construction in that in that context but um but anyway but this is a wide open question um so yeah interesting stuff the risk of asking a way to be okay i'm going to talk through this again i'm happy to explain this again then um no no problem okay so these diagrams okay that you see on the left they are not really kind of rigorous things these are more like mnemonics okay tools to help us sort of like you know so we can communicate with that during we know what we're talking about without being you know super explicit okay so the idea here is that you might want to construct a complex okay find a spectrum okay and that spectrum should have s mod beta the two cell complex as a sub complex so maybe i can write a little bit here right so we're looking for some some complex x it should have s mod beta it should receive a map from s mod beta and the quotient the co-fiber should be some sphere i'll just put an s star for a sphere okay that's one way of expressing the idea that um that s mod beta it can is the bottom two cells right and then there's a third cell s star okay and also there should be a map from a sphere into x and that that this one corresponds to the bottom cell here such that the quotient is the two cell complex s mod alpha okay you might ask for one x that has both of these properties okay and it turns out that they're that such an x exists if and only if alpha times beta equals zero okay and then the same idea pertains here maybe you know i don't quite know what to call it maybe we'll call it s mod gamma comma beta that's my name for the three cell complex okay should map into x and the quotient should be a sphere and then a sphere maps into x and the quotient should be s mod beta comma alpha okay um so that's a little more detail about exactly what this is now what this means right is somehow even more complicated right but you can kind of break it down into what the parts mean right and those parts have to overlap consistently and so forth all right and so i was gonna try sort of to finish this up here i should say that one x exists if and only if alpha beta is zero beta gamma is zero and zero is in the bracket alpha comma beta comma gamma uh and and so and what this the point here is that it's inevitable that you have to study total brackets the total brackets just simply are the answers to the questions that um that that we care about and so we we need to study them