 thank you very much for the invite to talk. I think it's Sam's talk he mentioned it was late in the UK, it's now significantly later in the UK. So this talk is going to be quite light. I will tell you a bit today about some types of functor calculus and how you can kind of move between these versions. So I guess the natural starting point is to talk about what a calculus actually is. This could be quite general but I'll tell you kind of the kisses that I'm interested in as we go. So we'll go back to our favorite thing from undergrad Taylor's theorem and we'll start off with the function say on the real line and you might ask what we can do with this and well one thing we know is that we can build it out of polynomial functions that are called Taylor polynomials. I just wanted to check really quick. Is your screen perhaps cut off a little bit? We see functor calculus and is that better? That's better. Yes, thank you. Okay so Taylor polynomials that are written like this and on the natural thing to do is just categorify all of these things right? So you want to replace a function you just make it a functor and we just take some functors from some category c and do some other category m and we replace these Taylor polynomials with essentially just polynomial approximations that I will tell you how to construct in a bit and we call those t of f and again there's one for every n and they behave exactly how you would guess that they behave. So you know something is n polynomial then it's n plus one polynomial. The n plus first derivative something n polynomial disappears so on so forth. So the benefit of Taylor's theorem is that these polynomials are in some way calculable you know you can just take the difference between the nth and the i minus one or i minus first and you get some formula involving the derivatives and a power of x and the natural question is well what happens if we try and do something like this in the functor calculus world and some difference here so once I have my category m up here is some kind of model category don't know what model category is just say base spaces every time I say model category and what you want to do is to come to the fiber of the map from the n minus first approximation or from the nth approximation to the n minus first approximation and well you can ask does this come with some nice description in terms of the derivatives and we'll we'll see that in a bit. So the benefit of this is that you can gather all these polynomial approximations together into a Taylor's power which you should just think of as the Taylor series right and what we're really interested in is one of the differences between each successive polynomial approximation and this whole thing sits under the the functor you want to approximate so we want to use information about these layers to tell us information about the polynomial bits to then get information about the original functor so it's all very abstract and a bit nonsense so what we care about particularly today are functors from the category of vector spaces over some field to base spaces and in particular only when f is the real numbers or complex numbers all right so when f is the real number this is called orthogonal calculus that was originally developed by Michael Vice in the 90s or the complex case is called duty calculus and it's been around for a while but hasn't really been well written down until last year so you might ask kind of an obvious question as to right okay you can do it but why should anyone actually care about this and so let's I want to give you a couple of the reasons as to why you should care so the first one is for the the home entropy theorists in the audience and I'm going to tell you a little bit about the stable home entropy groups of the sphere which is just the home entropy groups of the suspension spectrum of the sphere so if you know anything about home entropy groups and spheres you know that these are basically impossible to calculate but it's slightly easier to calculate than the actual home entropy groups the spheres so there is a a way to take strange enough information about the home entropy groups of spheres to tell you information about the stable home entropy groups it's called the EHP spectrum sequence and it has E1 page given by unstable home entropy groups and it converges to the stable home entropy groups of the zero sphere okay you probably have to put in some kind of too locally conditioned here but let's just gloss over that but there is another way that uses a type of functor calculus so in the 90s Goodwillie developed a calculus which looks at functors from base spaces to base spaces and also its spectra in these places and it also works fine in which case he showed that the home entropy fiber of the map between the anthroproximation and the animals first well we call that dm and it is actually given as an infinite loop space of some spectrum and the spectrum has an action of the symmetric grip that's not overly important for our purposes and this gives another spectral sequence called the Goodwillie spectral sequence with E1 page given by the home entropy groups of the spectrum let's say at some space x and it converges to let's change x to be a sphere and let's change it after we the identity functor because it's easier just say we have the identity functor on a sphere and it converges to the stable home entropy groups no it doesn't reverse the home entropy groups of the sphere so is p a dummy variable there then oh p should be a p yeah somewhere thanks that looks slightly better okay so the point is that this Goodwillie spectral sequence takes stable information in the terms of the this spectrum this dn or this dp and converges exactly to kind of the terms that are appearing in the php spectral sequence so in work by Barron he basically said okay these vector sequences inform each other so if you know something about differentials in one you know something about differentials in the other and vice versa so he did use this to do a bunch of different calculations of stable home entropy groups so hundreds of pages worth of calculations so where do vector spaces come in well that comes into work of Barnes and Eldred we saw similarities between Goodwillie calculus and their orthogonal calculus so they took a functor in the Goodwillie sense so one between base spaces and they assumed that this was nice then what they could show was that if you look at the Taylor tire that we constructed but this time in orthogonal calculus of the functor but you can pre-compose it with this thing that I would call s then that's the same I was looking at the tire in Goodwillie calculus of f I'm pre-composing that with s where s is just the functor that sends a vector space to its one point compactification so in essence what Barron was actually doing when he was looking at the Goodwillie spectral sequence on spheres was doing a orthogonal calculus on these on these spheres I will also say nice a bunch of times and every time I say nice it has a different meaning so just take that with pinch of salt so I'll stop here if anyone has any questions okay let's let's give another reason as to why you should care and this one comes in the form of some geometry and when you say geometry I'm coming at geometry from a stable home entropy theorist point of view so um pick the word geometry with a pinch of salt there is a theorem from the 80s of Miller and what Miller did was he looked at the space of linear isometries from cq into cm this is the steeple manifold of q frames in cm and he constructed a filtration on the space that was given or the quotients of which were tom spaces over grass manians and what you want to do is compare this space to its quotient so he took um the quotients of the filtration took all of them they're only q of them so it's not too bad I want you to compare these two spaces and I was able to show is that the steeple manifold splits after taking suspension vector so stably splits there are other filtration fun spaces that look like this so at around the same time so approximately in the 1980s Mitchell and rector gave filtration like a spell on very similar space you take loops on something slightly more general you take two vector spaces like the linear isometries between one and one plus another so linear isometries are always injective so we can always do this um so they give a filtration and it was in fact my whole old who made the following conjecture so he said well this comes with a filtration so something similar to the the Mitchell or to the Miller result should happen in that when you compare the space and the wedge of the quotients in the filtration you should well there are infinitely many of them this time but that's not too important again this should stably split and this should remain quite unsolved for quite a while so the whole made this conjecture 1980s 1990s it was early 2000s when it was resolved and it was solved by this point of view of looking at the space of linear isometries as a functor so the kind of point of view you want to take is that we can write linear isometries from some vector space into it plus something else as a functor which is the science of vector space to the to the right space of linear isometries and so applying loops to that still gives you a functor from vector spaces to spaces and it was exactly this point of view that let our own prove the conjecture from hold and actually just recover the Miller result as well so we took a functor from vector spaces over the complex numbers to base spaces with a again nice filtration let me call that fly and he showed that you recover the functor from the quotients again after taking suspension spectrum both sides and this exactly recovers the result Miller and the conjecture off the hold so now we're going to move on from why she care into how it works and try and tell you a little bit about how these polynomial bunkers are constructed and and then how we can compare polynomial things over or in polynomial things you ever see unless we have questions at this point all right so if you're familiar with um good really calculus at all you'll know that it relies a lot on cubical homotopy theory so something similar happens here right good really calculus really looks at cubes and spaces and properties of these highly highly dimensional tubes and spaces and what the functors do to these these cubes so let's start off by looking at a generalization of typical homotopy theory but what we call the orn cubes let's go back to the very beginning what is a square well the square is four spaces and some maps connecting them and what this translates to really is a functor from the power set of the set of two elements into space spaces so here to underline it's just set one two so the top vertex corresponds to value on the epicent the value on the one element section one value on the one element section two and value on the whole thing so that gives us a way to define what a cube is if you put in three here and draw the picture you'll get exactly what you think so you can keep going and define an n cube to just be a functor from the power set of the set of n elements to space spaces so the the natural guess is to want an orn cube as well we just go over to the set of n elements and replace it by the the vector space orn so we start off with the set of n elements and it gets transferred to the space vector space orn so what happens the power set well you look at what i'm going to call p of orn so this is the post set of subspaces of orn but importantly this has a has a topology so over on the the the cubicle side of things over here there's no topology everything's just discreet but over here there's actually an awful lot of topology going along and you need to really remember this topology at every step so we can use this to define an orn cube which you know is exactly what you think so an orn cube is a functor from this post set of subspaces of orn to space spaces so this is hard to imagine so let's let's draw some so let's start off with the easiest example an orn cube so the post set here you have a point that will correspond to your value on the zero dimensional vector space and a point that will correspond to the value one or and a line drawing them so this is no different to what you would expect a one cube to be but when you move to or two is when you see the real complication of the topology coming into play so you still have the zero dimensional vector space and its value you still have the value on the real line and then you're choosing an embedding of the real line into the plane you have a value in order to and you choose an embedding of the real line into the plane and you have to then in some ways keep track of the fact there are infinitely many embeddings right so if you're choosing to embed the real line in as say the horizontal axis then you have to keep track of all the rotations of the horizontal axis about the origin that's just a different embedding of the real line into the plane so that gives you an s1's worth of points here and you then have all possible lines connected so if you know much topology in these things this initial vertex is just the graph manian of zero dimensional subspaces of r2 this is the graph manian of one dimensional subspaces of r2 which is or p1 which is s1 and here you have the graph manian of two dimensional subspaces of r2 and of course i'm doing or here but put in C and it works just as well so in general this post set is what we call topological which is more than just being a category enriched in spaces it means it has it's a category that has a space of objects and a space of morphisms so the space of objects is actually quite nice right down so exactly what it looks like in this picture here where we have this district union of graph manians you just have all the graph manians below a certain degree do you think all the graph manians for k less than m of k dimensional subspaces of rm and the the morphisms of this space is also not horrific right down but i'm not going to do for the purposes of of time so what is the point well if you've played around with cubes before and cubicle homotopic theory there's a bunch of algebra going on here right you can talk about cubes being homotopic cartesian homotopic co-cartesian you know you can have a total fibers total co-fibers so on and so forth and the whole point is well they just came from the factory to finding this thing as a functor and we can do the exact same thing here we can talk with homotopic cartesian homotopic co-cartesian total fibers and so on so as an example of one of the definitions so a cube x or an orange cube x is homotopy cartesian it's the map from the cube up the initial vertex into the limit over the remaining vertices so we get rid of the initial of the cube is an equivalence so here all i'm doing is i'm taking the cube i'm getting rid of the initial vertex and puncturing the cube pulling all the vertices left back to get this limit and asking the map from the initial vertex to this limit as an equivalence and this is actually enough to define what a polynomial functor is so if you've ever read michael vice's original definition of this it is the exact same definition he just doesn't really say where it comes from and this is one point of view of where it comes from so we take a functor we can go back to being general because i don't want to draw pictures so from vector spaces over f again f is r or c okay and f is n polynomial so this meaning polynomial of degree less than or equal to n let's just click away running it if the map from f on some vector space v into the homotopy limit over the non-zero vertices the or n plus one cube is an equivalence so what we're saying is this is the exact same as taking your functor restricting it to an or n plus one cube such that when you take the functor look at the restricted and look at the initial vertex yes of f of v and you want this to be homotopy from teaching and this is for all v now this limit here is actually what we use to define then the polynomial approximations let me just call this tau n f of v so if you're familiar with goodwill calculus this tau n is goodwill is t n and my t n are goodwill is p n i didn't come up with the confusion but i'm just going to stick with it so we can define the polynomial approximation nth polynomial approximation of a functor to just be at the stabilization of this this tau n functor so you take the limit of the co-limit over iterated applications of this tau n view as a functor and this gives you the universal n polynomial thing so i mean it's a bit of work too it's even n polynomial but it is and if you have a map write it from f to e where this e is n polynomial this will factor through the n polynomial approximation but just up to homotopy so are there any questions at this point what is the reason it does not suffice to take tau n once hey it's not n polynomial okay so you might be thinking right okay we can do this but these or m cubes are pretty horrific to actually do anything with are there any calculations so let me give you a calculation when i say calculation i'll write it i won't do it so there's a functor which at v you send to the stabilization of the real projected space which v would add this to our base point this is in fact one polynomial so to see this you can play around with a bunch of different cubes and figure it out or what you can actually do is see that f is the first approximation of the functor tolerate as this which sends a vector space to the quotient of the infinite orthogonal group what are the orthogonal group in that vector space and you calculate this directly from the Taylor tire which i will try and tell you a bit now so we saw before that we had this tire we want to know what the the error between this successive polynomial approximations was so this has a similar format to how it appears in the giggly calculus so if you know what that is and so this is originally due to my advice for the case when f is or and myself when f is complex numbers so we had these errors of our approximations and some vector space v what you get is you get an infinite loop space again if it's a spectrum this spectrum is given by you take the derivatives i haven't told you about but they exist and you can make this into a spectrum with an action of on or un you suspend it a bunch of times depending on your vector space and the approximation you're looking at this smash product has a bunch or has a un action or no one action so you mod out by the action and take the infinite loop spaces resulting spectrum where should i explain what some of these things are so this theta fn is just the spectrum with an action of odd n and odd n is just the odd morphisms of fn so it's either on or un and nb is just n multiples of v so we get exactly what we would expect if you know good with calculus you get a you get a version of the same result any questions if not we can then move on to how we can compare these two okay so let's take a take a bit of a pause on the calculus for a minute okay and let's look at some topological k theory so if someone asks you to give them topological k theory you're probably going to tell them by k u right which is the complex version which is of course a real version just replacing where you get by complex vector bundles with real vector bundles and there are maps that go between them and on the level of spectrum these are even even rig maps so there's a complexification map which i'll call c so you just on the vector bundle level you just tensor over or with c and there's a map in the opposite direction which i'll call or and this is what's called realification or a de-complexification and all you just forget the complex structure so or of some ck is just or 2k to underlying set and only a lie multiplication by the real numbers so k was built from real vector spaces so there's kind of a an ad hoc analogy between it and the orthogonal version of the calculus and the same thing we said then on the unitary side so k built from complex vector spaces so with unitary calculus and there's an ad hoc relationship here kind of an analogy and the question is is there something sitting between these two versions of calculus that make this diagram of analogies commute i mean is there some version of this realification complexification relationship happening on the level of calculus and is it good for anything so the answer actually turns up yes that's pretty much exactly what we think it would be so if you go to the level of vector spaces so vector spaces over or and vector spaces over c this is a complexification and realification form an adjoint pair so we can use these to build or to move a functor from taking values in vector spaces of raw or taking values in vector spaces over c so if we start say with the functor that starts off in vector spaces over or i want to move it to a function that starts off in vector spaces over c well there's kind of only one thing you can do and now let's just add in the map for vector spaces over c vector spaces over or this realification map and then you just take the composite which is pre-compositioned by or of course the same thing then works if you start off with the other type of functor and take the other comparison so we start off with a functor e between vector spaces of diversity and spaces and we just add on to the start this complexification functor and take the composite so the natural question is okay is it good for anything does it actually preserve any of the things she would want it to preserve so i mean the existence of these is essentially trivial but i mean if it doesn't stand to the n polynomial approximations to the n polynomial approximations is it good for anything the answer is no but luckily in our case it does actually do what you'd want to do so how are we going to do this is we're going to work backwards along the tire we're going to start it off with the errors i'll work backwards to gain information about the polynomial functors and then the the entire picture so the layers of the tire are actually a special case of a more general type of functor so a functor is what we call n-homogeneous if it is both n polynomial and it's n minus first approximation bashes so the layers an example in fact the theorem we had before was classified the layers of these even if this is a spectra worked for any homogeneous functor if not just the layers so the starting point as well what do the comparisons do on these homogeneous things you should think of these as things that just have an n polynomial part but no no lower polynomial parts floating about so what happens on the homogeneous parts is the following so we start off with something n homogeneous in our orthogonal calculus so from now on f is always going to live in our orthogonal calculus and you will always live in the integer calculus you don't have to keep writing down them as functors start off with something n homogeneous and after this verification we get something again that is n homogeneous and in fact what might look weird initially you start off with something n homogeneous in unitary calculus, complexify you get something that is actually 2n homogeneous so the 2n seems a bit weird to start off with that's absolutely what you should expect so if you realify a complex vector space you double the dimension if you complexify a complex vector space or a real vector space it preserves dimension but because we're pre-composing with both of these things you should expect the direction to swap so pre-complexifying with double dimension and pre-realifying with maintain dimension but what's annoying is that this doesn't actually tell us anything about the layers so unfortunately for any f and then e there's no clear connection between the layers but we can look at a special type of functor for which it works so let's do it in the f case so if f is a different version of the word nice then taking layers of f and then moving it in the calculus or looking at areas of f moved from orthogonal calculus in unitary calculus is the same thing there's there's no difference and the same works then if we take something in unitary calculus and apply this pre-composition with the complexification under this condition that f is nice so this nice condition I can spell out a bit what this means and this just means that the map from f and v into its polynomial approximation is let's hope I get this right take the dimension of the vector space plus the dimension of the polynomial thing plus one I like to shift it by a constant connected and this is for for all in so okay so these polynomial approximation things are notoriously hard to calculate so do we actually know any functions for which this works so that's say if you start writing down a list of things you would want this to work for it turns out to work for them all so examples of nice functors so we have this sphere functor from four which maps a vector space to its one point compactification you look at the functor bo which maps a real vector space to the classifying space of its orthogonal group you can look at the unitary version the complex vector space goes to classifying space of its unitary group and you can look at things like representable functors which just sends a vector space to the linear isometry from some fixed vector space to all of these are all of these are nice and the even better thing about these nice functors is that their powers converge to the original input functor so just by coming up with this definition of nice we have a bunch of of new calculations in this so then we can just extend this along to the polynomial parts just by using the fact that this dn is the home of your fiber between the map between the nth approximation and the n minus first approximation so I'll just give the the the good version so if f is nice then it doesn't matter if you send f into the unitary calculus and then polynomial approximate or polynomial approximate and then send the approximation into unitary calculus and then in the case when you're looking at something in unitary calculus you just get a two n appearing in the right place so this is moderately annoying right you have this this condition in which works for the functors you want it to work forward but somehow it doesn't seem to work for them all and in particular I should say that this all implies that when f is nice it doesn't matter when you look at the Taylor-Tar of f as a functor in unitary calculus or the Taylor-Tar of f in orthogonal calculus and then pre-composed with this realization functor hey Nile yeah I think your screen might be getting cut off again a little bit can we see it now well I see it so like um I see like if f is nice then yeah sweet yeah is it still there yeah all right so we have this nice thing and we want to try and fix it in some way so it brings me on to kind of my last point of how we can fix this so um I should say at this point things become joint work um joint work with Greg around from Stockholm and so to fix this what we do is we introduce a new meaning of dice and but I'll give it its actual words so functor f is analytic if when you restrict f to an orion cube with kind of the same condition we had before on the initial vertex of the restricted cube so the initial vertex should just be f up some vector space you want this to be another level of Cartesian but this time it's kind of a weird looking formula involving a shift of the dimension of the cube on another shift okay with these r and c is just constants that can be anything and so the point is that these analytic functors are essentially the building blocks for any functor in say orthogonal calculus so we could replace or would see again and would work but so what I mean by this is that the building blocks is that if you want to construct a cellular approximation to these functors in the projective model structures and I'm saying words that you don't understand just ignore me so if you want to construct a cellular approximation after this functor in the projective model structure well the the cells that you use in this projective model structure are all analytic you can show that they preserve different unions and pushouts and so forth so that the approximation will be a functor will be approximated by a sequence of analytic functors so in the this actually means is that any f can be written as a filtered hundred column of some alphas with each alpha analytic and once it's good for well in particular uh analytic implies the nice from before so if you have an analytic functor and pre-composing this verification functor preserves the polynomial approximations so you can get that uh for any f doesn't matter if you polynomial approximate an unitary calculus or a polynomial approximate an orthogonal and then shift the calculus so in particular um we get the statement for the tires we get an equivalence between the tire and unitary calculus and the tire and orthogonal calculus then shifted to unitary calculus so are there any questions at this point does equivalence always mean point wise view equivalence yes okay so i have five minutes so i will say one more thing in these five minutes and that takes me back to topological k theory so we've had ko and ku but they have a kind of bigger cousin that tells you all you need to know about ko and ku so if we remember a diagram from before we had ko complexified we got ke about the structure got ko kind of sitting above both of these is what's denoted ko and it's what's called k theory with reality and so this is uh constructed by a tier and what it is it's the unitary or the complex k theory you take into account that complex practices have an action uh of c2 by just complex conjugation and the thing was that ko and taking c2 fixed points completely recovers ko and forgetting the c2 action completely recovers ko so what you can ask is does there exist a calculus version of ko or of ko sorry and the fact is that there does so you can think about what's called calculus with reality as construction is kind of the obvious thing you take the unitary calculus and put a c2 action on all the vector spaces everywhere you can think of putting it okay and so this means that instead of looking at bunkers from that land in spaces we have to look at bunkers that land in spaces with an action of c2 so it's functors from vector spaces over c but remembering that they have complex conjugation to c2 spaces and this whole construction works just as you think it would work so we can even classify the layers in this in this case so in this calculus from the reality we get essentially the same classification as we had for unitary calculus but this spectrum that we built from the derivatives is actually not a spectrum with an action of the semi-direct product of un c2 so here you get c2 act on un by complex conjugation of matrices and kind of the big picture on this whole calculus story is that if you start off with a functor from vector spaces over c with this c2 action to c2 spaces you can get a functor in orthogonal calculus well by the way you would think you add on a vector spaces over or at the start and you can classify to get a vector space over c2 and this has this or overseeing this has this c2 action and then to get to from c2 spaces to spaces you take homotopy c2 fixed points and then you just take this functor and you can do then the the kind of other obvious thing starting with the same functor well you can just kind of ignore the c2 action on the vector space variable and forget the c2 action that we end up with and take this functor and the point in the final minute is that this so meaning these things you see might completely recover the orthogonal unitary calculus all right and I'll stop there all right thank you now so it's it is four o'clock but um let's maybe take a minute or two for questions real quick so does anyone have any questions can you clarify what what you precisely mean by completely recover these two calculus so the unitary version is the easiest one so you take a functor in this calculus with reality and that is equivalent to taking a functor in unitary calculus you get an equivalent of categories pretty much so you have to forget some action but when you forget the action there's an equivalent of categories and one functor and the functor is polynomial in the calculus with reality if and only if it's polynomial in the unitary calculus after you forget the action and then you get the same kind of statements for the orthogonal version okay thanks so the the categories are really equivalent as categories are only up to a month of years and so there's a there's a subtle change of universe thing that has to happen and that gives you an equivalence of categories