 Thank you. Thank you very much for the invitation to speak. I'm really excited to be able to participate I'll be it remotely. I'm glad that that worked out Less glad that it had to work out kind of wishing that the coronavirus were better under control especially here in the US but Yeah Since I'm given the last talk I I get to to also say I think we should thank the Organizers, I mean that they've done a wonderful job taking an An in-person conference and very quickly turning it into a great online conference So please join me. I guess somehow Yeah, thank you Ian and great job Guys, this is seriously fantastic, and I know it was a lot of work. So thank you very much Okay, so as I said, um, I'm I'm gonna talk about techniques of computation in Equivariant and and then some in motivicomotopy Trying to hit on as many of the key words from that tidal of the conference as possible I'm I'm aiming this to be more Introductory than sort of the research side and please do if you have questions Ask away, and I'll try to answer them as we go through Most of my focus in the talk is going to be on the Equivariant computations a lot of them for for Everyone's maybe second favorite group the group with two elements And the reason I'm going to be focusing on this one is is Well several fold First the group with two elements, which I'll call C2 from now on is the Galois group The complex numbers over the reels this ties it to a lot of Geometric and algebra geometric concepts namely if I want to talk about say descent for real vector bundles It's the same thing as understanding a complex vector bundle together with this descent data of a C2 action Second a lot of classical and chromatic Computations can be seen in this C2 Equivariant story and actually I know that that you all saw Dan Isaacson's series of talks earlier in the conference where he talked a lot about the connections between Motivic over R, Motivic over C, C2 Equivariant homotopy and and classical homotopy So I'll pick up on some of those themes as well And then finally and this one really I should have led with because in some ways it's the most important from my perspective We can actually do computations here a lot of the literature about Equivariate homotopy theory tends to suggest that computations are essentially impossible Some of which go so far as to say it's impossible to do some of these I I don't find that to be the case and I hope that by the end of my talk You agree that a lot of these computations are much more doable Then you may have thought initially Okay, so I'm going to start by just saying that we're going to be working in the following context There we go, blossom a bit of a spell. I'm going to be working for Working in what people sometimes called genuine Equivariate homotopy now I I I Hate the word genuine here. It's for two reasons. It's very value laden Especially when you compare it to the the contrast we will talk about naive Equivariate or genuine Equivariate. There's a distinct hierarchy Established there and it's not actually supported in the math so what I would advocate for and I hope I can start to get traction in this is Not to call this genuine, but rather to call it something instead like complete and the complete here means that we have All transfers And this is a theme that I'm going to spend a little bit of time talking about as we go forward But before I do that, I'm actually going to start with just a little bit of a review so how do we talk about Computations and where do the invariance live? So how do we Understand how do we understand? Homotopy groups G spectra and G spaces and Dan also So sorry there is a comment by Yuri Suleyman He says that one version of nice just G spectra is often called borel complete. Yeah, that's true I don't understand the frowny face there Yuri It is the case that that the homotopically meaningful version of naive G spectra is the is borel spectra and here We don't necessarily have transfers. These are things we sort of free up the action historically And I guess the not having transfers is is probably why you had the frowny face So I'm with you on that one So in the equity context the first thing that I run into is I can't get away from thinking about homotopy sheaves instead of homotopy groups and Of course the real way we should be doing algebraic topology and sort of this ideal world where we have complete control over everything Is I would be able to just immediately tell you what maps out of any say finite CW complex were I'd love to be able to tell you what maps out of any finite CW complex are We approximate this instead by restricting attention to maps out of the building blocks of finite CW complexes Namely spheres. I'm going to do the same thing for for G spaces or G spectra So we'll consider we'll look at Functors and these are functors from the category which I'll call fin G op in to say a billion groups and This fin G op. This is the category of finite G sets and Equivariate maps and I'm going to do this just via The yonata lemma and so the ones I'm going to care about are I'm going to take equivariate homotopy classes of maps from Oops, I'm describing a functor from t plus so t together with the district base point smashed with the end sphere Into some fixed G Spectrum E and my superscript to G here is just reminding myself that I'm taking the collection of equivariate maps this is this is a Contra variant functor as written because I'm mapping out of the T plus slot and so in particular it fits into this form and this is what I'm called the Homotopy coefficient system Already, I'm suggesting a way that I should be thinking about my G CW complexes the CW complexes I'm going to build not just out of spheres with a trivial action, but rather out of spheres Again with a trivial action But I allow myself to to take disjoint unions of these and to permute the copies of the spheres in those stacks And that's how the group is going to be acting So I can map out of this and this amounts to picking out maps from spheres into various fixed points So the first thing to notice Is that if I have a district union of things so T district union T prime and then I take the Distant base point and smash this with the end sphere map this into E Then the inclusions of T and T prime into the district union give me a pair of maps Backwards So I get a Decomposition whoops like this So in other words my functor from finite g sets up into Abelian groups isn't any old functor It's one that takes the district union, which is the co-product in finite g sets Which makes it the product in finite g sets up to the product in Abelian groups So in other words this construction gives me a product preserving Functor and I want to stress here that I'm in the algebraic context So saying that I'm a product preserving functor is a property of the functor rather than additional structure Well any g set has an orbit decomposition. So this functor slot plus smash SN into E is determined by the values on Orbits by which I mean Transitive g sets So g mod H as H varies over the subgroups And if you haven't seen this before then I would Suggest that you spend a little bit of time thinking about what what's the geometric content? Maps out of g mod H Equivariate maps out of g mod H into some g space and you can start to see the interplay between fixed points for various subgroups and then g itself Okay, now this is the kind of thing that I didn't need to be working in G spectra. Yeah, I could have made sense of this in G spaces provided n was at least two and If I'm in g spectra any kind of g spectra be it the Burrell ones That Yuri brought up be it the complete ones that I'll be working in or anything in between I still have these homotopy coefficient systems the key feature of the this complete Equivariate homotopy is that I have not only these Contravariant restriction maps, but also the covariant transfer maps so I'm going to define a category the Burnside category of g has objects finite g sets and the morphisms So home in the Burnside category from s to t is going to be Well, I'll be a little glib here and just put parenthetically the group completion of the set of Correspondences s and t so here I have two Equivariate maps f and g and then I'm doing this up to isomorphism So again, I'm doing I'm going to be working in an algebraic context so I'm working up to isomorphism as as you know from Clark's work or Angelica's then I could have instead considered an enrichment of this be it one where I have a two category and Instead of considering this up to isomorphism I remember the isomorphisms as the two categorical part of the data or an infinity category where I build much larger Diagrams again recording isomorphisms and various pullback conditions Oh, and I should say if I'm going to say that I have a category I need to say what the composition law is and composition is via pullback so given two Correspondences, then I can pull them back and I get another correspondence Okay, so In this category since the category is the same as that category of Excuse me the objects are the same as the category of finite g sets I can still talk about things like district union and Cartesian product In this category though Well, the Burnside category it's Canonically self-dual and by canonically I mean it's the identity on Objects and then I just observe that I have my correspondence Which has you know that maps in two different directions and that's just a sort of an artifact of the way I'm writing I'm choosing to read from left to right because That's the only way I know how to read English, but I could have instead swapped it and Gone from right to left then I would be seeing instead Hom from as written right here. That would be the same thing as com from t to s So a is oh, I should have given this a name Sorry script a so a is Canonically self-dual and the district union is Now both the product and the co-product And if you haven't spent any time working with this category sort of thinking through what this might look like I would suggest seeing for yourself how the district union could possibly be the product in other words see How do I write down maps in the Burnside category from t district union t prime back to t and back to t prime? Whereas in general, I'm not going to have those maps just in finite g sets So the the players in the equivariate context in this complete one Are Mackie functors? these are so a Mackie functor is a again product preserving Functor from the Burnside category into Abelian groups and I'm always going to indicate my Mackie functors with an underline So that there'll be a little bit of type checking to contrast these with Abelian groups Again any g set can be decomposed into orbits and if I use that orbit decomposition I get that a Mackie functor is determined by a much smaller amount of data So let me spell that out For G is CP and I'm going to name a generator CP. So choose a generator to be say Gamma Okay, a CP Mackie functor is the following data of first an Abelian group M of g mod g which is a point second I have a CP module M of CP and Third I have maps a Restriction Which goes from M of a point To M of CP and to transfer M of CP To M of a point and I'll often write this as a little diagram That that people call a Lewis diagram after Gauntz Lewis My restriction goes like this And my transfer goes up and then I have an action of My group CP on the on the CP module And then these satisfy a few axioms so first the restriction The restriction lands In the CP fixed points and the transfer Factors through The co-invariance and then second I Have a condition called the Mackie double coset formula Which says that the composite of the restriction with the transfer is The sum over the elements of the group which is items called the trace If you use the gawa theory names Okay, so that's it and how am I supposed to connect these two Correspondences, so how am I supposed to see this as something coming from the Burnside category? Well, remember that I have in the Burnside category. I have Or excuse me in the category of finite CP sets. I have a map CP to a point That's just the crush everything map And I have a map from CP to itself. That's multiplication by gamma And this gives me a little commutative diagram because point is terminal Now when I think the category of finite g sets, I can embed that co-variantly as the forward direction map in the Burnside category or I can embed it Contravariantly as the backwards map in the Burnside category and if I embed it Contravariantly that gave me my restriction map This one and if I embed it co-variantly that gave me the transfer map So both of these two maps here the restriction and the transfer arose from this quotient map from CP to a point and Then the first conditions this one about the image of the restriction landing in the fixed points or the transfer factoring through the orbits are Exactly summed up in the commutativity of this little triangle. So it's it's actually just the functor reality condition And finally this Mackie double coset condition. This is what you see if you pull back CP over a point with CP over a point and the pullback is CP cross CP and then I want to write that in terms of CP sets so I want to break it up into its orbit decomposition and when you do that you get exactly this condition Okay, let me make it a little more concrete Because I will actually do a couple of computations later and I want to be able to to use these so There's the There's the representable functor the burnside Mackie functor the value at a point is given by Z direct sum Z and the value at CP is Z and Then the restriction and transfer maps I'll just write them as little as little matrices this one sends the first thing To one and a second to P and the second is Zero one The vial action here is Just by the identity So as I said, this is actually the the Functor I get by mapping out of a point in In the burnside category again, this is product preserving because Distant Union was the product and so it's it's that it's literally the universal property of the product to say that harm Out of a point is product preserving Okay, if you've also seen the burnside ring the burnside ring is the Groten deep group of finite g sets so I should also be able to connect these two summands to finite g sets and I can this summand is the g set Point and the district union of copies of point this one is the g set CP as A CP set and every CP set breaks up into a district union of points and CP and then my restriction map is just forget the CP action and just remember the set and That takes point to a set with one element and it takes CP to a set with P elements and that was this map Okay, so the other that I want is the constant Mackey Functor Z and this one is The value at point is Z The value at the at CP is also Z the restriction map is the identity The viral group whoops, sorry While group action is also the identity and then that forces the transfer to be multiplication by P Because since the restriction is injective then I can compute the transfer by computing the composite of the restriction with the transfer And I see I have no choice here The These two Mackey Functors are pretty closely connected The target is what sometimes called a co-homological Mackey Functor Now at this one, I should pause and connect this already to what we see in the motivic story Often when we talk about pre-seves with transfers in motivic homotopy, we're referring to things like this that are close to co-homologic Mackey Functors and There I see the same kind of condition that the composite of the transfer and the restriction is multiplication by the index of the group and That's the condition that I'm writing down here In a covariant homotopy we allow these more general kinds of transfers Which you should think of as actually also showing up in the motivic context This is analogous to the transfer along say a finite at all now, okay So Before I continue questions about this so far. I Know a lot of this is review But that doesn't mean the questions won't have come up you said you said a and the line is a is a Bernstein Mackey Functor. Yes. Yes. Okay, and it's the usual thing in math where proper names become Become adjectives and so you end up with long strings So Sean asks why co-homological is this an important distinction These do these do show up a lot and and they're the kinds of of Mackey Functors that you see with group co-homology and That's that's one of the reasons why I might Describe them that way So from that perspective they it is a very natural class of Mackey Functors that arises So there's been a lot of work in this for us The the constant Mackey Functor Z is a fairly easy one to do computations with as I'll as I'll show you in just a minute And it also arises naturally in the Equivariate context Yes group co-homology does take values in these Group co-homology naturally has an extension to a Mackey Functor and And when I do group co-homology, I consider it in in one of these contexts and they're always coming from Modules it's always something that's a module over the constant Mackey Functor Z No, thanks for asking okay, so The reason that we talk about Mackey Functors in Equivariate homotopy is that Mackey Functors play the role of Abelian groups in in Genuine G-Spectrum in other words all of our usual algebraic invariance Actually Mackey Functor value So for example normally I might talk about Homotopy groups of a spectrum and In the Equivariate context, I have the homotopy Mackey Functors An Equivariate spectrum I can talk about the a Generalized co-homology theories value on a space or Spectrum and in the Equivariate context, I have a Mackey Functor It's worth of the co-homology of X in some E theory So I have a richer structure that I could be working with For those who might Worry or wonder about such things that they're the category of Mackey Functors is an Abelian category We have enough projectives and injectives so we can do homological algebra the way we normally would and then In a little bit. I'll also talk about how the category Mackey Functors has a symmetric minoidal product So we were really exactly like with Abelian groups as Reflecting what we saw in spectra. We build a model in DG Abelian groups In we can do the same thing in Equivariate spectra. We take Equivariate spectra and we compare it to to DG Mackey Functors or DGAs in Mackey Functors So one of the things that I want to be able to do is talk about ordinary Homology and I find it easier when I'm talking about ordinary ordinary homology to just show you how to compute this in some examples so How do we compute homology? Remember, this is supposed to be a Mackey Functor, but I'll just tell you the value of this At at some point with coefficients in something and now just for simplicity for myself. I'll start at this point switching to the group being C2 So I Had I had I planned I Had to use some of that newer technology. I would ask via a poll. What's your favorite way to compute? ordinary homology You know, it's like we would do in a calculus class do like a quick spot check But if I were to do that, I would guess I would guess you would say cellular as opposed to singular although singular is certainly nice for sort of Formal reasons. Yeah, thank you. And yeah cellular is the way that we actually can compute things easily We write down a small chain complex do it so let's do that here and let's start with an example, so We'll just do this via cellular homology and my example is going to be Let's look at a representation sphere So I'm going to take S to the C by which I mean the one point compactification of C and Then C remember I said earlier My C2 is also the Galois group of C over R And so this this naturally has an action of C2 as the as the Galois action So if I were to to draw this well, this is the this is the Riemann sphere So I have the real line sitting inside the Riemann sphere and then I have the two hemispheres And here was my S to the R Sitting as the equator and S to the R. Well, this is just S1 and When I think of the two Hemispheres in my Riemann sphere Well, I could put them in as As showing up and actually I'm doing a different projection than you're probably thinking of I'm gonna have the positive Complex part being the upper hemisphere excuse me the the positive imaginary part being the upper hemisphere and the negative imaginary part being The lower hemisphere so my group acts by swapping the two hemispheres and Leaving the equator fixed So I can build this as an equivalent cell complex. I have two copies of The one sphere and they're swapped. So I'm gonna have a C2 cross S1 Because that's two copies of the one sphere I'll draw a cartoon as I go through. It's my one sphere and my one sphere and they should have been the same and the group acts by swapping them This is a C2 cross it and I'm gonna map this to the one sphere Where I just fold them down. So it's via the identity. It's a twisted version of the fold map And then I can include these into the corresponding C2 cross discs and that amounts to just putting in a little disc on each of these and When I pushed this out So actually let me Oops When I push this out Now I've exactly built my Riemann sphere With the two hemispheres that are swapped so here's my cell structure and If I want to take the cellular homology, well, what I need to do is figure out What am I supposed to do when I evaluate? What's the homology of one of these G mod H? cross a sphere or G mod H plus smash a sphere So the building block is I'm going to take the homology Hn of Whoops, you know, so H star of G mod H cross An end sphere with coefficients in some Mackie functor M Well, this is going to be I'll do reduced. This is zero if star is not n and it's just evaluate M at G mod H If star equals n Now I can start to write down what what my homology is going to look like notice that this map here this one this is the same thing as C2 to a point Crossed with the one sphere and remember my Mackie functors are exactly built so that they know what I'm supposed to do to Maps between Orbits so a map C2 to a point. This is something that I can evaluate my Mackie functor on Now I can write down the chain complex using that so in degree zero again, I'm doing the reduced theory I have nothing Here's my degree in degree one Well, I had my one cell. There's only one one cell and it's the one sphere So I have M of a point And in degree two I had a single Equivariate to sell It was the one coming from M of C2 and the cellular boundary map is just M of C2 going to a point So notice this is the covariate version of this. So this is the transfer From M of C2 to M of a point And this tells me how I can write down the this Homology for any of these so I get H1 is the co-curnal of the transfer and H2 is The kernel of the transfer and it's a little more work. You can get the These as instead Mackie value to things it amounts to thinking about G mod H and and putting in another slot Where I crossed with some fixed T okay So since the this is a talk in the broader context of a summer school, maybe I'll say as an exercise for you whoops as an exercise Figure out homology groups of S K times C with coefficients in any Mackie functor M For all K and M you use the same idea that I talked about here You have to think a little bit about what happens in the co-homology version So namely when K is negative, but it's a it's fun to to work through Okay There's one other thing that I want to point out here and that's actually right here. I Am going to give a name to this map From S1 into this C sphere. I'm going to call this a sub Sigma I'm going to call this sometimes the Euler class of The sign representation and if I'm being super pedantic, I'd actually call this the suspension of a sigma because a sigma is a map from the zero sphere into Now instead, it's just the one-point compactification of that imaginary axis in C Where that's swapped because well, we know how complex conjugation works and here I'm seeing this co-fiber sequence C2 goes to the zero sphere or C2 plus goes to the zero sphere and the co-fiber is this sine sphere And this whole part that I'm writing down in this case is the suspension of that co-fiber sequence So it's something to keep in mind all right, so I I brought this up because Just doing these computations Understanding the co-homology of these spheres the K times C spheres as K varies Allows you to get a lot of mileage equivariately, so I'll start with a theorem And this is due to lots of people Individually, I'm gonna say that the first parts of this is due to Duggar It's due to who crease and it's due to me Hopkins and Ravana, and that is that there's a filtration on The C2 spectrum of real boredism So M. You are or Again, if you're coming from the motivic context You should think of this as MGL and then the theorem is due to different people so in motive motivically, it's due to Hopkins and Morel and We're walk and there's a filtration on M. You are Ding with associated graded gr of M. You are is I'll just write it as the Eilenberg McLean spectrum Associated to that constant Mackie functor Z and then I'm putting in a bunch of formal indeterminance where the degree of each of these indeterminants is Just like in in dance talks my indeterminance are going to be graded by representations or in this case, they're by graded actually have two irreducible representations And this is just I times C or again. I'm using the complex conjugation action on C What this means is if I want to compute the M. You are homology or M. You are co-homology of some space or spectrum then I have a spectral sequence and the e2 term Is given by well say the homology, but I'll write it this way I'm going to do the homotopy again homotopy Mackie functors of the function spectrum From say X into Each Z I join these indeterminants Well, this is just the homology oops, sorry the negative co-homology of X with coefficients in Z and then I join a bunch of it determinants and this spectral sequence converges to the M. You are co-homology of X So it's like in a tea. Here's a brook spectral sequence But I'm using this different filtration and the filtration is the slice filtration Named after the motivic slice filtration Waivotsky that Was was done by dugger initially Okay, so what I want to focus on and I'm seeing that Time quickly passes. So what I want to focus on is that this is a spectral sequence of Mackie functors So this is sort of the the first Order approximation to understanding the way I can do equivariate computations Mackie functors form an abelian category Which means I can talk about spectral sequences of Mackie functors And in this case, what does that mean? We have two spectral sequences There's the fixed points The value at a point And there's the underlying And then they're connected. I have a map of spectral sequences That's reflecting my restriction map from the fixed ones the underlying And I have a map of spectral sequences from the underlying back to the fixed points And the underlying was actually a spectral sequence of c2 modules So I have all of this added structure that comes in It's it's a lot of added structure But it's not an insurmountable amount of added structure The biggest thing I can do is I can use the fact that since these are maps of spectral sequences If I have a class that's a cycle or maybe a permanent cycle Then the image of that under any map of spectral sequences is a cycle or a permanent cycle If I have a class that's the target of a differential Then I know that under a map of spectral sequences It's still the target of a differential. So I get a lot of additional constraints on this So as a as just an example In the in the spectral sequence Computing the homotopy Mackie functors of M. You are The ideal generated by two is an ideal Of permanent cycles and so why it's just because Two times any class x or remember I'm looking at something where I started with the constant Mackie functor Z and in the constant Mackie functor z two was Was the transfer of one in the underlying And then I have this Frobenius reciprocity condition. Let's me move the x inside. This is the transfer of one times the restriction of x So this is the transfer the restriction of x And in the underlying spectral sequence the underlying spectral sequence is just the ordinary attia Here's a book spectral sequence computing the homotopy of mu out of the homotopy of mu So it it collapses with no extensions And the restriction here Is a permanent cycle Oops, it's not how you spell always So that's giving me a huge amount of information about the structure of the spectral sequence that I don't know how I would have known otherwise I needed that this This large number of classes namely twice anything could actually be written via the Mackie structure as As a permanent cycle Okay, so in the time remaining I need to push into bigger groups And I need to talk a little bit about the the The added structure. I've already started to dance around some of this first You'll notice I used a different wild card here than my asterisk I used a five pointed star here here Here and then here I just used the ordinary asterisk. So this one this wild card was Following notation of who increased. This is the roc2 grading So I actually have more information That I have at my fingertips And second I talked about an ideal here Which says that I should be thinking about this actually as a spectral sequence of rings And that's true, but I won't go too much into it. So in fact The slice spectral sequence is a spectral sequence of commutative monoids in Mackie functors And these are called green functors Sean asks If there's a that there's a result saying differentials are power operations Yes, yes The Maybe the the best way to say what the differentials are In the classical a t here's a spectral sequence is that they're cohomology operations Because they're maps connecting between Onlandberg-McLean spectra and here in for For something like m u r the The fibers are again suspensions of onlandberg-McLean spectra. So the initial differentials are exactly Co-homology operations in this case from Co-homology with constant z coefficients to itself and then all of the higher differentials can be expressed as secondary or or higher order operations Just as we would see with the ordinary a t here's a brick spectral sequence In this case, it's a it's a consequence of knowing the form of the spectral sequence knowing that the fibers are all These generalized onlandberg-McLean spectra But yeah, I can't think of them in exactly that way Okay So I'm In that the time remaining and I've said that already I want to do one last added bit of structure So here I've used that the mackey structure shows up and it gives me a way to produce a bunch of permanent cycles And to transport differentials Then I know that this is a spectral sequence of these ring objects these green functors So I understand that at each page I have a ring and for each g mod h I have a ring the restriction maps are all ring maps And so I can use all of this to To continue to bind classes to other classes and simplify the problem The last part is to use the norm so we have Also multiplicative transfers And these are actually arising from functors quite generally on on the complete Spectra, so I have a norm functor from h spectra To g spectra And this is a symmetric minoidal functor That's going to take some e and you should think about it as going to I'm going to smash together G mod h copies of e i.e This is a tensor induction And these norm maps have the property that since the tensor product is the co-product on commutative rings I have canonical maps We have canonical maps for any Commutative ring In g spectra I have a map from the norm Of the restriction of r Back to r this endows the Homotopy Mackey functors of r with these external norm maps And here I have to use the grading by the representation grid So just as earlier when I talked about the sum over the Vial group or the sum over g being the trace I'm using the galwa theoretic language there here. I'm also using the galwa theoretic language You should think of this as being as being heuristically the product over over g mod h Of some element And since my ring is commutative If the group is acting by permuting now the tensor factors around but the multiplication is actually commutative So it doesn't care what order that they were in this gives me a way to take an element that's fixed by h and produce an element That's fixed now by g The this structure was first studied by Tambara who looked at these and called them t and r functors these sort of mackey functors together with multiplicative transfers And the last result is the slice spectral sequence Is a spectral sequence of tambara functors This one I don't know how to show this and in the Motivate context the analogous operations the analogous norms and normed motivic spectra were done by Bachman and hoi wa they described How you can think about the norm maps and how to build these added norm External norm maps on commutative monoids in this context And I would expect that the slice spectra's even should have this property In the equivariant context The somewhat surprising feature is that the slice filtration is actually the universal filtration That has the property that it takes commutative monoids in spectra to a spectral sequence of of Tambara functors. Um, and if there are questions about that afterwards, I'll answer it So I I I saw you pop in Frederic, so I know that I'm That I am almost out of time. So let me just say a punch line And that is uh, what the hell does it mean to be a spectral sequence of tambara functors? So means that we have so I was popping for questions, but you can you can take times Oh, okay. I reserve my question all right So we have a twisted Version of the Leibniz rule and let me just spell that out in one case So I have the I have that the differential On some class that I write as the norm Of x again Churistically, so I'll do this in a different color because this part's a lie This is supposed to be x times the conjugate of x And then I know from the Leibniz rule how to compute the differential on a product. This is the differential on x times gamma x plus x times the differential on gamma x Remember the differentials were maps of mackey functors so I can pull The vial action out So this is dx Times gamma x plus x times gamma of dx And this is the same thing then as one plus gamma on x times oops not the one I want to do dx times gamma x And now I can make this true statement, which is the transfer of dx Times gamma x So this is the the version of the Leibniz rule that shows up in this case The differential on the norm. It's just like on a product But I'm supposed to remember that the norm was a kind of product where the group permuted the factors around And so the differential is going to take that to a sum where again the The group permutes now the summands around And that's exactly the role of the transfer You can do better though This is saying something just about d of n of the norm. It's just the usual Leibniz property And so this is the very last thing I'll say And You can do better if d n Of x so now I'm in my slice spectral sequence is y Then I actually get a longer differential d 2n minus 1 of That same classroom before a sigma times the norm of x is The norm of y For this, I don't know a classical antecedent. This is saying that instead That once I put in this a sigma it actually the differential almost behaves like a ring map at the expense of shearing it from That d n to d 2n minus 1 So it's like saying If I know the differential on x then I can is y Then there's some kind of differential on x squared that looks like some kind of y squared and these these Collections of properties I've described them are the way That people are doing computations with these spectral sequences So I think I'll stop there Okay, so first my thanks a lot for the nice talk And so we We are so let's fire the first question So can you read it or do I read it? Yeah, so Sean asks couldn't I view a sigma n and n is two different power operations? And then it would look like some of bruner's work. Yeah, I think that's exactly the way that I want to do this They it's I should be able to connect a sigma n to some kind of of actually in this case Diolashov operation because I'm looking at an operation on homology um, but I don't I don't quite know how to make those work. I'd love to talk to you more about that um Yes, they Sean also asked do these norms prolonged the category of filtered equivalent spectra. Yes, they do um Uh, Yuri asks is the universal property of the slice filtration written down anywhere? No, um Yes, I don't think Maybe but I don't I don't recall Oh, wait, maybe in the handbook of homotopy in the chapter. I wrote for the handbook of homotopy I believe I talked about the universal property of the slice filtration There so thank you for Make me remember that and then um Anonymous attendee asks could I mention some of the applications to chromatic homotopy has promised in the abstract? also, yes So the Applications of some of this And I'll be quick The applications are first. Let me recall a theorem of Han Chi And this says that the lubin tate spectra En for any n are real orientable in other words, I have a map Of ring objects in the homotopy category M. You are to en knowing this if n is two to the k minus one times anything so times m Then the Hopkins Miller theorem says that c two to the k acts on En Which gives me then via just the that the Norm forget a junction I described above a map from c2 to c2 to the k of m you are into En That's again a map of ring objects, but now in the homotopy category of c2 to the k spectrum And then recent work of of me She and ming kong jeng Says that you can use this to build a model for e theory en as the kn localization of Of some quotient of and you are And to spell out exactly what stuff is would take me a little far afield But the important thing is the slice spectral sequence Here has a describable A more understandable He too and then et cetera terms Then the corresponding lube and tape theory had we knew the c2 action On en but being able to describe the c2 to the n action on en in a way that we could write down the homotopy fixed point spectral sequence that was that was sort of the the Bloody edge of the state of the art using these sorts of Equivariate methods and stepping through the norms and these sort of quotients of the norms of mu The slice e2 term is very easy to write down and to describe And then you can use the techniques that I was describing over the course of the talk to Sort of bootstrap differentials inducting up over the order of the grew And use this to get a lot of information about the homotopy groups of Of the hopkins miller spectra In ways that we never were able to before Okay, so so I have a question What's the link between tambora and green functons? So There's a forgetful functor every tambora functor has an underlying green functor So a tambora functor you can think of as a green functor together with these additional multiplicative norm maps And then there's actually Just as there was a hierarchy That started with coefficient systems and it ended in mackey functors where I start to put in more and more transfers There's a hierarchy between green functors Which are tambora functors with no multiplicative transfers all the way up to tambora functors which have all multiplicative transfers And this hierarchy I can it's exactly analogous to the additive hierarchy for the mackey functor case and This is this is an important Feature, so thank you for bringing it up Zorisky localization does not work well In tambora functors. So for equivariate commutative rings Zorisky localization doesn't preserve the property of being a commutative ring It does always preserve the property though of being sort of the spectral version of a green functor so an algebra over an e infinity operat But it's it's a very particular kind of e infinity operat one in which the group doesn't act And so that's a a subtlety that shows up and makes some of the computations a little trickier Okay And also I have a kind of maybe vague or broader question so So you describe mackey functor and they are defined for finite groups, but is there a theory for other groups? So first example would be profile night groups like yeah Yeah, um, yes, um, and There are several several versions of these for the um The cases that were most studied in in classical homotopy theory were compact lee groups Where again we have a good notion and they're and you have mackey functors for a compact lee group and they're describing the homotopy groups of a a genuine g spectrum for g compact lee There the multiplicative version of these only shows up for finite index so pairs of finite index subgroups There's no sort of degree shifting part that can show up in the in the compact lee for profile night Dresden's even ochre have a a bit vector construction for For sort of profile night groups that's generalizing the ordinary bit vectors and they're Describing the profile night version of the burn side ring in that case because the bit vectors The bit vectors of z is is where the the truncated bit vectors are exactly giving me the various burn side rings as I look for Like c and or whatever my truncation system was Barwick also in his spectral mackey functors has a really beautiful approach to understanding the profile night case of a mackey functors as well Beyond this you can nothing nothing that I was writing down really depended on On the group being finite. Um, I could still talk about finite G sets for g not finite I start to run into pathologies like if g is divisible there aren't any interesting finite g sets And then I'm going to start to run into trouble but aside from from those cases You can talk then about You can talk about mackey functors. You can do all the same thing Last question. I was also thinking about I don't know if you know this The rust cycle modules. So it looks really like mackey functors, but there are two operations that are added So it's kind of multiplication by unit and residue maps for valuations and it It it made it could met you we could see that as mackey functors for the so-called motivic galois groups where you have transcendental extension So I've you seen something like that and I guess Um in I haven't but but that's a That's an interesting thing. I'll think about that I think I mean one of the reasons that I wanted question Yeah, yeah One of the reasons I wanted to give this talk is I think a lot of the techniques that we've been using in the Recently in the equi variant context should port through without change Into the motivic one all the stuff that we've been seeing with the multiplicative transfers anything showing up in the in the bakman hoi wa Normed motivic spectra. We should have analogs of these two kinds of conditions on differentials in certain spectral sequences And this last one the one that changes degree It's allowing you to lift differentials multiplicatively in a way that's That can actually be pretty surprising To get new ones. So I'd love to see the analog of that Motivically okay So it seems we have no more questions. So again, Mike, thanks a lot for a nice talk