 Thank you, Dennis, and thank you for inviting me here. This is actually my first visit to IHES. And it's very nice here. So since I was already busted once, I gave this talk at the Clay Math Conference in the fall. So if you were there, you've heard this before. And I apologize. Anyway, this is kind of an evolving project with a lot of aspects that Arvind Azok and Jean Fazel and I are working on about approaching the problem of constructing algebraic vector bundles using methods of Mahomet-Tapi theory, and especially approaching the problem of trying to give a topological vector bundle an algebraic structure. So the context will be varieties just over the complex numbers. And the basic question I have in mind is which complex vector bundles have algebraic structures, like I just said. Now that is a really hard problem. That's a positive solution to that would imply the Hodge conjectures, even just in the case of affine varieties. So I'm not going to talk about this problem in general, but I'll eventually focus on the kinds of varieties that are as far away from the Hodge conjectures you can get, where all the integral cohomology is of Hodge type, and hopefully say some interesting things about that. So this story and this interaction really goes all the way back to Sear's conjecture and to Sear's FAC. Oh, I don't know why that background got so dark. And in that paper, Sear's exploring this relationship between vector bundles on a space and finitely generated projective modules over the ring of functions on the space. So if there was a topological space, those would be continuous functions, and it was an algebraic variety or an affine algebraic variety. That's just the ring of algebraic functions. Sorry, something happened. Well, I apologize that that background is so dark. I just changed it, and it changed back. Anyway, so in that paper in FAC, Sear writes, it's not known if there's any, when X is just affine R space, if there's any finitely generated projective modules which are not free. And of course, he's thinking by analogy with topology where vector bundles over a contractable space are all trivial. And this became known, of course, as Sear's problem, even though it was just a question, and eventually it became known as Sear's conjecture. I guess Sear has that kind of karma. And that problem was solved independently in the 70s by Quillen and Seusslin. So the theorem was exactly that, that if K is a field, every finitely generated projective module over K is free. Just give me one second. I don't know why that is so dark. Oh, it'll lighten up in a minute. All right, so these are just going to be dark. OK, so as we said, it was solved independently by Quillen and Seusslin in this case. All right, and so there was a whole bunch of problems like that, and many of them promoted by Hyman Bass and Sear and others. And Adams kind of identified, in his math review of these papers, he identified an ongoing program. This shouldn't imply that this was Adams' program, but he articulated it in this math review, saying, well, I know you can read that, but he says it leads to the following program, take definitions, constructions, and theorems from bundle theory, and express them in terms of finitely generated modules over a ring, and then use bundle theory as a conjecture-generating device and try to prove these things about rings. And one of the sort of maybe one of the most definitive conjectures along these lines is what's known as the Bass-Quillen conjecture, and that concerns. So I'm going to be looking at vector bundles in various categories. And so I'll just write Vect K for the set of isomorphism classes of rank K vector bundles and then decorate with algebraic, topological, or whatever. And so the basic thing about topological vector bundles is their homotopy invariant. The set of isomorphism classes of a vector bundle over a space across an interval is the same as the isomorphism classes of vector bundles over the space. And the Bass-Quillen conjecture asks that that be true if that's true for algebraic vector bundles. And I think you need some hypothesis. And I think the most general form of the conjecture I know of is that when A is a regular ring of finite curl dimension, then every finitely generated projective module is extended from A in the sense that I wrote. So that would be the exact analog of the thing in topology that vector bundles are homotopy invariant. So as far as I know, as well, this problem isn't solved in general. But in 1981, just building on the Quillen-Sussland proof Lindl showed that the Bass-Quillen conjecture was true when A is finitely generated over a field. So that's called the geometric case. And if you're interested in this and you don't know about it, TY Lamb has a wonderful book on this Sarah's Problem on Projective Modules. So this leads, if we really take Adam's articulation of this program seriously, it leads to this question, can we study algebraic vector bundles using homotopy theory? That was sort of a lot of conjectures and theorems were setting up the basics of doing that. But what would happen if you really tried to go the distance and really try to use the methods of homotopy theory to study algebraic vector bundles? Ah, so now abstract homotopy theory comes in and lightens everything up. So in order to do that, we have to have an abstract framework for talking about homotopy theory. And so I mean, this has come up in two of the three talks already today, but I'll also review it a little bit. So in the setup for abstract, so abstract homotopy theory, I mean, it probably really also starts with Groton-Deek, but it sort of starts to take on an apparatus of definitions in the work of Dan Quillen and Dan Kahn. Quillen, of course, had found himself doing homotopy theory in many different contexts, and he wanted to prove that different approaches to the same homotopy theory were the same. He wanted to prove that differential graded algebras modeled topological spaces up to rational equivalents and differential graded Lie algebras and co-algebras and all that, and he needed an abstract framework for making sense of that. And Dan Kahn spent most of his working career kind of boiling this notion down to its essence. And nowadays, at least in algebraic topology, we think of the basic arena of abstract homotopy theory as just consisting of a category and a subcategory of morphisms that you intend to identify as equivalences. So the language is the terminology is borrowed from homotopy theory, but you're really just localizing, or in some sophisticated sense, just the way Seher localized Abelian categories. So in this language, you have a category and a subcategory of weak equivalences, and you consider functors called homotopy functors that take the weak equivalences to isomorphisms. And there's a universal homotopy functor that's characterized by what the, well, it's almost the obvious universal property. Given a homotopy functor, there's a unique functor making the diagram commute. If you're kind of thinking about categories or higher categories, this might seem like kind of a rigid thing to request because you're characterizing an arrow in a two category, and so it should only be characterized up to a contractual groupoid or something. But the difference between this rigid thing and perhaps the more complicated but natural higher categorical statement is just the one of asking that the objects of the homotopy category be the same as the objects of the category. Anyway, that was just me editorializing. It's not going to play an important role. There's a universal homotopy functor. I keep losing this. So the classical example is topological spaces and the class of weak equivalences are the ones that induce isomorphisms of homotopy groups at all possible choices of base points. And that becomes classical homotopy theory and the homotopy category is the usual homotopy category of CW complexes. So I'm mentioning that just to kind of locate the origins of the terminology for you, but also to introduce some notation. Nobody really in their right mind ever writes out that symbol. And usually it's abbreviated with square brackets. So whenever in all these worlds context where I'm going to talk about abstract homotopy theory, I'll just use square brackets. And some subscript to indicate the context. All right, so homological algebra also gets absorbed in this abstract homotopy theory world. In fact, Quillen called abstract homotopy theory homotopical algebra. Oops, I got ahead of myself. But maybe it doesn't matter. There you take the category of chain complexes, possibly bounded below. And the weak equivalences are quasi-isomorphisms. And your intention is to regard, is to only study functors that send quasi-isomorphisms to isomorphisms. So as far as doing homotopy theory with algebraic varieties goes, I think the first place that's set up and kind of the frameworks laid down is in the thesis of Ken Brown. I didn't put a date down, but that I think also goes back to the late 1970s. And his thesis was published in a paper called Abstract Homotopy Theory in Generalized Sheaf Co-Homology. And really the main thing there is he wanted to set up a world where sheaf co-homology was homotopy classes of maps into some kind of an Eilenberg-McLean space suitably defined, and to be able to talk about generalized co-homology like we do in homotopy theory. And so this isn't precisely the category he set up, but it's convenient, and it connects better with the ones I want to talk about. So I want to consider the category of smooth varieties over the complex number. And I'm sorry, I mean, that's the basic. Grotendeck topology, or that's the basic category. And I want to just consider the category as simplicial pre-sheaves on that, so contravariant functors to simplicial sets. And so that doesn't have anything to do with recovering. I mean, you want to recover some sort of local, the global principle. You want to recover the sense in which your smooth variety was built out of smaller pieces. And so you don't impose that relation in a sheaf condition. You add that to the weak equivalences. So the weak equivalences are the original simplicial weak equivalences. And then for a technical reason, which plays a role throughout the talk but won't play a role at the level of detail I'm giving, you use the Nisnevich topology. So the weak equivalences are the Nisnevich hypercoverings, which expresses the way in which local things accumulate to global things. And the simplicial weak equivalences, which was the whole point of putting in the simplicial pre-sheaves. So this leads to what I'll call algebraic homotopy theory. And I'll decorate that with an alge. And in this world, you can formally put in a number of McLean objects for any sheaf, or pre-sheaf even, of Abelian groups and calculate hypercohomology or sheaf cohomology in terms of the homotopy category. Okay. And then there's Motivic homotopy theory. So that's basically the same, except you take the algebraic weak equivalences and you force the affine line to be contractable. So we take all, now we wanna add to the weak equivalences all those maps, the projection maps from the product of any x with the affine line to x. And so that's Motivic homotopy theory. So an invariant of algebreic homotopy, studying homotopy functors on the algebreic category would just be things like sheaf cohomology and systematically studying homotopy functors on the Motivic category would be studying things that are A1 homotopy invariant in a systematic way. So there's what you might call realization functors. The Motivic equivalences contained the algebreic equivalences. So for formal reasons, there's a functor from algebreic to Motivic homotopy theory and over the complex numbers, since the affine line is contractable in topology, there's also a realization functor to topology. I might have maybe a better notation would have been C upper and because it's the underlying analytic. It sends a variety to the underlying analytic variety. Okay, so that's the basic setup of abstract homotopy theory. And I said I wanna, the question was, can we study vector bundles in that using abstract homotopy theory? Oh, sorry, we're back in the dark again. This'll, whatever. All right, so in, yeah. So I really, I thought I was to change all of them to light. I didn't really mean to say, only the discussion of abstract homotopy theory is gonna have a light background, but hopefully that'll, hopefully that'll go back. Okay, so in topology, if you have a group, there's a classifying space for principal G bundles and Siegel introduced this simplicial definition, which I think is probably familiar to everybody. And in topology, principal G bundles up to isomorphism is given by homotopy classes of maps into BG. So the problem of writing down the set of isomorphism classes of K-dimensional vector bundles is the problem of determining the homotopy classes of maps into some classifying space. And the same thing is true for formal reasons in the Ken Brown algebraic homotopy theory. I'll say more about that in a minute, but that really just unpacks to the local description of vector bundles in terms of charts and transition functions. And so you also have a statement like that. There's a universal vector bundle and vector bundles becomes a homotopy problem, but homotopy doesn't involve deformations parameterized by the affine line. So anyway, with that in mind, let's just abstractly define motivic vector bundles to be the set of maps in the motivic homotopy category from X to BG okay. So right now that's just an abstract definition and there's the realization maps between those categories give me a map from the set of isomorphism classes of algebraic vector bundles to the one the isomorphism classes of motivic vector bundles and then to topological vector bundles. Okay, and so one can study the problem of take the problem of giving a vector bundle on a topological on the analytic variety underlying a complex projective variety. What could take the problem of finding an algebraic structure on it and ostensibly factor it into two separate problems. But in fact, there's a good relationship between these that, so let me turn to. So the first thing is there's a theorem of Morrell. So Morrell's published his theorem in 2012, but I think it goes back quite a bit before that. And then Marco Schlickling and then later Arvind Asakmark, Oywa and Matthias Wendt found a much simpler and more direct proof of this theorem. But the theorem says that for smooth affine varieties, this map, the map from algebraic and motivic vector bundles are the same thing. And you can think of that as a sort of really definitive version of Lindel's theorem or even the Quillen theorem that Lindel's theorem saying that at least, at least this has a chance. Lindel's theorem says that algebraic vector bundles is a one invariant. And anyway, this is kind of a, it's an improvement of Lindel's theorem and in some sense the more definitive statement that you might wish for. Okay. Okay, so that's the case of smooth affine varieties for projective varieties. There isn't really much of a chance. Projective, the set of isomorphism classes of algebraic vector bundles on projective varieties isn't even deformation invariant. So I wrote down an example here, but which is sort of pretty elementary to say, but just take, if you take like take O of one, O of one, we'll take the Euler sequence on P one. So that writes O of one plus O of minus one. Or yeah, as an extension involving, I'm sorry, that writes the trivial bundle as an element and as an extension of O of one by O of minus one. And now just move the X class in a straight line to the origin. I just wrote down an explicit way to do that. And that gives you a deformation from O of one plus O of minus one to the trivial bundle. So probably everyone in this room knows that kind of thing very well. Anyway, so that set is not, is not a one, one invariant. And so there's sort of no hope really to describe, it doesn't, I mean, it's sort of a non-starter. You're not gonna describe algebraic vector bundles in terms of motivic homotopy theory. However, so this is a question I don't really know the answer to. I don't really expect this to have a positive answer, but I don't really know the answer. And that is, the most naive answer would be, that's all you do, the set of motivic vector bundles is just algebraic vector bundles, modulo, modulo the equivalence relation generated by identifying a vector bundle over x cross a one with its fibers over zero and one. This seems quite strong to me, but I don't know, I don't really know a counter example to it. And you'll see this, this would have a lot of, so this would imply for instance that every motivic vector bundle had an algebraic structure. And again, I don't really know a counter example to that either, and in some sense that's gonna be the main thing I'm talking about. But anyway, I'm putting it up as a naive guess. I don't really expect that to be true, but it would be nice to even know to understand this quite a bit better. Okay, okay, so nevertheless, there's something you can do by exploiting Joannou's device. So I just, so here's the easiest example I know of to explain it, take affine n space minus the origin. So that's defined by algebraic inequalities and that's not a affine variety. But if you give an algebraic reason for the x's to not be zero, so if there's a, if you take this ring, you add some y's so that that sum is one, then that is an affine variety and that's a torsor for a vector bundle over affine n space minus the origin and so it's a weak equivalence. And then if I mod it out, the evident action of the multiplicative group, then I would get a bundle of affine spaces over projective space and in fact, and then you can restrict a closed sub, in fact there's a quite general story here, but if you have a quasi-projective variety, there's always an affine space bundle where this j of x is an affine scheme and that j of x is equivalent to x in the Motivic-Homotopy category because those affine spaces are contractible and it's also smooth if x is smooth because it's a zirisky locally trivial bundle. Okay, so anyway, the upside is every smooth variety is weakly equivalent to a smooth affine variety. And so the problem of whether a Motivic-Vector bundle has an algebraic structure is equivalent to a purely algebraic problem because Motivic-Vector bundles on x are equivalent, or the same as Motivic-Vector bundles on the Juwanalu device and those are the same by the theorem of Morel and Azok and Oiwan went to vector bundles over the Juwanalu device and so the question I was asking is sort of really comes down to the question of whether vector bundles on Juwanalu devices extend to the base space. And I don't know, I don't know an example of one that doesn't, it's easy to find vector bundles can descend in many, many ways, and anyway, it would be nice to have hands on an example of one that doesn't descend, but. Okay, all right, so that's a little bit of a tour of the easy relationship or sort of the known relationships between algebraic, Motivic, and Motivic-Vector bundles. And so now when I say I wanna study vector bundles doing homotopy theory, what do I have in mind? Well, so in any abstract homotopy theory, anytime you set it up, it comes to you with its own kind of internal notion of homotopy groups and its own kind of internal notion of cohomology. And there's always, there's a steamrod kind of obstruction theory you can always, that sort of also always, you can always exploit and it always relates the ith cohomology of X with coefficients in the ith homotopy group of, or that's supposed to be an X to maps in the homotopy category from X to Y. So in algebraic homotopy theory, if we were to look, so this obstruction theory gives you an idea of what these categories, how these categories are kind of digging into the theory of vector bundles. And so this gives you another way of seeing why the algebraic homotopy theory is, it's just a formal, it's really just kind of formal. The homotopy groups or the homotopy sheaves of BGLK are just GLK when I is one and they're trivial otherwise. So this just becomes the statement, the obstruction theory just becomes the statement that vector bundles is H1 with GLK coefficients. So there's nothing that doesn't penetrate into the theory of vector bundles at all. On the other hand, motivic homotopy theory does. So the fundamental group of BGLK and motivic homotopy theory is GL1. And pi two is the second Milnerwit, un-ramified Milnerwit K theory. And there's a little bit of information known about the higher homotopy sheaves, but it's a little complicated and it does lead to an interesting obstruction theory. I'm sorry, pi two and BGLK? Very well. Pi two. Oh, pi two of BGLK, yes, thank you. I'm sorry, yeah, thank you. Yes, not two, right? Yeah, and I probably need K to be at least three or something. Yeah, sorry about that, thank you. So I wanna talk about this obstruction theory for a minute, but so this obstruction theory, this will apply if you're studying vector bundles over smooth affine varieties. And that was something that was studied pretty intensively by Griffiths and others in the 1970s using analytic methods. So Griffiths idea was you take the under, the complex vector bundle by Grower's theorem has a unique holomorphic structure. And if you can give the, and so you have a holomorphic connection and if you can make that connection algebraic, you have an algebraic structure. And so he studied using value distribution theory, the growth rate of the connection form as you go to infinity. And he found kind of a series of obstructions to algebra, algebraicizing vector bundles. And these give you a different series of obstructions that are very homotopy theoretic and very kind of discreet feeling. And one of the things we haven't really worked out, but I think would be really interesting is to understand the relationship between the co-homology with coefficients in these homotopy sheaves and the value distribution theory that Griffiths was looking at. Cause they're not, and there's a kind of, there's a, anyway, they're not obviously related to each other at all. So Morel and Azach and Fazl investigated problems of splitting free summands off of projective modules over regular rings in low crawl dimension using these methods. And Jean and Arvin and I, so this was sort of more a proof of concept at this point than anything. Cause, so the other thing from that era in the 70s was a sort of philosophy I don't want to call it really a, there was just a sense that if the churn classes of a vector bundle were algebraic, the bundle was algebraic. And that's probably because that program was designed to get at the Hodge conjecture and the idea was once you, all they wanted was the churn classes to be algebraic. But so we sort of dug into this and just produced an example of a smooth hyper surface in P1 cross P3 for which the complement has algebraic churn classes, but it's not algebraizable. And I think this would be, we haven't done it for this example, but I think this would be a really interesting one to see why from the point of view of value distribution theory that this bundle, this rank two bundle doesn't have a whole more algebraic structure. Is it stably algebraizable? That is, if you make it the rank big enough. If you make the, yeah, yeah, yeah, yeah. This is important that it's two dimensional. If I make the rank bigger, it'll become algebraizable, I think. I think that's right. Yeah, I think that's right. Is that what your question was? Yeah. Okay, so as I said, I really, what I think, so that's a little bit of a tour of kind of one thing that's going on, but where we're starting to see some interesting project is progress is in kind of the kinds of varieties that are as far from the Hodge conjecture as you can be. So that's the problem of defining algebraic vector bundles on projective space. And line bundles, we know, so the first interesting case, first interesting problem is to classify the rank two bundles on projective space. And that's, of course, a very famous problem. There's Hartzhorn's conjecture, which says that if n is greater than five, every rank two bundle is a sum of line bundles. And there's something, I'm gonna mention it because it plays a role in the story I'm telling. So I don't know what to call this. I'm gonna call it the Grower-Schneider problem. And it says that every unstable rank two bundle, unstable in the Mumford sense, every unstable rank two vector bundle on Pn is a sum of line bundles. So that's for n four and on. And I'm calling it a problem sort of the, well, because it was a paper and then shortly after the paper was published, there was a mistake found and the proof has never been repaired. But it impacts, the timing of that paper impacts the story I'm telling. Okay, so if the churn class is a zero, that implies the vector bundle's unstable. Now, so in the 1970s, so this Hartzhorn conjecture was round, the Grower-Schneider theorem was a theorem. And so then the question turned to some topologists, produce some topological vector bundles with no churn classes, for instance. And so Larry Smith did it, Elmer Ries did it, and Bob Switzer kind of made a elaborate table of vector bundles, topological vector bundles on projective space through a range of dimensions. And I think these are all ranked two bundles. So Elmer Ries, using Homotopy theory, constructed some ranked two vector bundles on Pn with no churn classes. And at the time of writing, that was those were examples of topological vector bundles with no algebraic structure. And, but now the current state of the yard is there's still no known example of a topological vector bundle on projective space that doesn't have an algebraic structure. And one question we might ask is, do the, you know, is this, was that wrong? Do the Ries bundles actually have algebraic structures? Now, okay, so let me tell you a little bit about the Ries bundles. So in topology, you study the Homotopy classes of maps into BU2 in terms of the co-homology of projective space with coefficients in the Homotopy groups of BU2. And those are by the long exact sequence, or the fact that, you know, whatever, those are the Homotopy groups one dimension down of U2, or SU2, or S3. So what you meet when you try to understand ranked two topological vector bundles on complex projective space are the odd Homotopy groups of the three sphere. Now the first odd Homotopy group of the three sphere, if at the prime P, is in this dimension pi four P minus three. And Elmer Ries just took that element and made the obvious vector bundle. So you take this complex projective space of that dimension two P minus one, collapse out the lower dimensional projective space to get a four P minus two sphere and go over by this first, by a chosen generator of that group. And then Elmer Ries proved that these were non-trivial for all P. He determined to what larger projective space they could extend. He answered the kind of questions you would wanna know about these. And those bundles, of course, have no churn classes, at least for P, something three maybe, and bigger because they vanish on the complex projective plane inside there. So that's predicted to have no algebraic structure. Okay, so those are predicted. And so now I wanna go back to this sort of way I was setting this up and ask do these bundles have motivic structures and if they do, can we get those to have algebraic structures? And so in order to do that, we need to kind of see if we can do the same homotop, the same kind of, we need to study the same obstruction theory in motivic, the motivic homotopy category. So in motivic homotopy theory, there's not just an S-N for every N, there's an SPQ, and I learned this stuff from Morrell and Wojcicki kind of in the late 90s and was taught to use these two indices. So the first one stands for dimension and the second one is kind of what you can think of as the hodge weight. Some people nowadays, and if you're one of them, forgive me, some people use a different convention on those. So the one one sphere is GM or the affine line minus the origin and the one zero sphere, which you should think of as really just combinatorial is the affine line mod zero one. And since the affine line's contractible, I mean, that's really just the circle in the simplicial direction. And more generally, SPQ is gotten by taking the smash product of these with each other. But the way these come up in more geometric questions is the affine N space minus the origin that has dimension two N minus one and there's a little bit of hodge weight of N in there and N dimensional projective space mod, oops, that's supposed to be a PN minus one, that's the two N sphere. Okay, so then there's motivic homotopy groups and there's a topological realization functor as before and under that, these migrated spheres, they just go to the underlying topological sphere of that dimension. And so there's a map from pi A, B in motivic homotopy theory to just the ordinary group pi A. So what you're expecting when you look at this, the way you expect, so a lot of familiar elements in the homotopy groups of spheres lift to motivic homotopy theory. The pi N of SN is Z, you can get the hopf maps, but then there's a lot of more elaborate discussion things in homotopy theory where you're using the fact that some product of things is zero and you're building some higher order product and there's room in there for the hodge weights, for there not to be a consistent way of lifting the hodge weights. So what you're looking for when you're trying to show that something fails to lift from ordinary topology to motivic homotopy theory, sort of the only thing you have to look for is some, it's, I mean, I think of it as kind of a elaborate refinement of the hodge conditions and it's somehow, I don't know a great way of expressing it, but what you're looking for is there not to be a consistent way of putting a hodge weight in to some construction. Now, I actually had prepared, I wanted to show you a construction of this element in pi four p minus three, but I can do that on the board later, but I decided not to do it. I've done it, it doesn't, I found it, it doesn't go so well in this format, but if you want to know about it, I'll tell you about it, but I want to tell you one deep theorem that goes into it. So, you have to get started somewhere and where you start in homotopy theory is with the element in pi two n of bu, the generator for the k theory of the two n sphere. So that by instability comes from an element in pi two, pi two n of bu n or it's an element in pi two n minus one of u n. You start with that element and that element has degree n minus one factorial when you map u n down to the n sphere. So this is the key. You start with that and then you do a bunch of things. And I just, the one, so there's some tricks, some localization, there's sort of some boilerplate stuff in homotopy theory that you can imitate in Motivic homotopy theory, but the key thing that gets you going is this map from this odd sphere into the unitary group. And that's provided in algebra by this beautiful theorem that was in Seussland's thesis called the n factorial theorem. And that says that if you have a unimodular row, a sequence of elements of a ring that could be the first row of an n by n matrix with determinant one, no, you don't get to have it be the first row of a matrix with determinant one, but if you raise the entries to enough powers where the products of the powers is n minus one factorial, then there does exist a unimodular matrix. And that's what gives you this element, this lift of this element in pi two n minus one of u n. That's what makes that algebraic. So this mismatch between n factorial and n minus one factorial is just because I started counting with one instead of zero. Okay, so anyway, you can use this, you can use this secret construction, which I'm willing to tell you about, that I haven't told you about it. You can use this construction. And if you follow that, you wind up with a Motivic lift of this first element of order p in pi four p minus three of the two sphere. So that has order p, it's got dimension pi four p minus three, and it's got hodge weight two p. Now, if we go back and look at what we wanted, we wanted to imitate Reese's construction by modding out a lower dimensional projective space, and that gives us, well, so the dimensions, trust me, they have to work out the, the number to pay attention to is the hodge weight. The hodge weight was supposed to be two p minus one, and we got something with hodge weight two p. So the hodge weight is wrong, and you might think that there would be a way of exploiting this to show that these don't lift to Motivic vector bundles. So the theorem is they do, they lift to Motivic vector bundles, and the reason is this element had finite order, and in Motivic homotopy theory, there's this operator row that lowers, you can lower the hodge weight on the torsion in the homotopy groups. And it's pretty easy to explain what that is. So the, so let's take the, let's take that, I'm gonna take the degree n map on S one one and I'm gonna cone it off. So that mn is the more space for the mod, the degree n map, or it's the mapping cone of the degree n map on S one one. And I can, so I'm over the complex numbers, so I can choose a, just take a linear map in A one that connects one to a chosen nth root of unity. And that gives me a map from the other sphere, S one zero into this m one one. Okay, so those, so that's an operator, right? If I have an element in pi one one, or higher of order n, I can compose it with this and get an element in pi one zero. And these are compatible as n varies. And what is this realized to in topology? So that circles wrapping n times around the origin and that line is my line from A one mod zero one in. And in topology, that's of course homotopic to just that arc which goes once around the bottom circle. So in homotopy theory, this row realizes to the standard inclusion that just says, so what operator is that? That starts with an element of order n and says, oh, that I didn't just, it was just, it was an element. I don't care anymore that it had order n. It's just restricting. So a map from the one thing is as an element and a no homotopy of n times it. And I'm just forgetting the no homotopy. Okay, so that gives you this operation realizing to the identity map. And that lets us lift the RISC bundles to motivic vector bundles. So, so this is something, so these are rank two projective modules over that ring I wrote down, over the degree zero part of that ring I wrote down. I have no idea how to write those down as projective modules. We know that if you add a free module to them, they become free of rank three, but I don't know, we can't go through every step of the construction and produce those. And in fact, every vector bundle, every topological vector bundle, I know from Switzer's chart from there, I can lift, we can lift to vector bundles over the Juwanalu device. So this method gives you, it produces a lot of new rank two modules over these projective modules over this Juwanalu device. The problem of descending from the Juwanalu device to projective space, that could be where that seems hard. And I don't have anything good to report on that. It could well be that we've traded one hard problem in for a different one, but these, but nevertheless, there's a lot of new rank two projective modules over these rings. So that's the idea. So that's the main result, is that at least the Reese bundles, and there's a lot of new, motivic vector bundles on projective space. But that wasn't really the main, that was part of the main thing I wanted to talk about. So I told you that where you're expecting something to go wrong in lifting something from ordinary homotopy theory to motivic homotopy theory is that the hodge weights start to sort of, there starts to be something inconsistent about them. But there's this row operator. And as soon as stuff starts to have finite order, you can shift the hodge weight down as much as you want. And this makes one wonder if there's a way, a sort of general idea that might show that you can always lift things in certain cases from topological things to motivic homotopy theory. And there is, so this is a kind of hypothesis. I guess I would even think of calling it a conjecture called the Wilson space hypothesis. And it takes something that I think is one of the best kept secrets in homotopy theory. And the conjecture is that the analog holds in motivic homotopy theory. So I'm just gonna take a couple of minutes and describe it. So in homotopy theory, there's this remarkable class of spaces, we call even spaces. So they're even in the sense that their odd homotopy groups are zero and they only have even dimensional cells. Now, that's simple. There's a whole bunch of really non-obvious things that come out of this. So for one, you know, it turns out that the even homotopy groups have to be torsion-free. So if the space has finite type, those have to be free abelian. That's not obvious from the definition. They turn out to be infinite loop spaces. And in fact, there's prime ones. There's a Wilson space, an even space that starts, there's sort of a minimal one for every even sphere. And I believe those have unique infinite loop space structures and everything is built out of those. This theory, it has a remarkable rigidity. So examples of these spaces are infinite dimensional complex projective space. That's a KZ2, BU, the classifying space for the infinite unitary group and BSU. So I wanna just point out, we know these are even, the homotopy groups we calculate for some reason, the cell decomposition comes from algebraic geometry. I mean, we have algebraic cell decompositions for CP infinity, BU and I suppose BSU. So the first example where you don't know anything like that is the fiber of the universal second churn class. So in topology, that's called BU angle bracket six, but it's not a very useful, I mean, that's a notation that tells you what it is, but it's not suggestive of anything else. But that space has only even dimensional cells and I don't know an algebraic geometry reason for that. I'm sorry? It is even, yeah. Yeah, that's the first, yeah, that's even. It follows from the theory that it's not from a, just from a, just looking at flat varieties for BU. So, it comes from what? How do you, what are you asking? How do we know that's even? You have, you just, so what goes into that is knowing the Seier spectral sequence and knowing the co-homology of the Eilenberg-McLean space and the action of the, use the Steenrod algebra, the Eilenberg-McLean space, et cetera. So I'm about to ask the same question in Motivic homotopy theory where we don't know, I mean, knowing that's even would tell you the co-homology of KZ4. You can go back and forth. So it's, it's, it uses that. But how do you know that even, is it the first or the second condition that you have? So it has even homotopy groups, that's obvious, right? It has even cells because I calculate its co-homology and I find it's free Abelian and only in even degrees. So in topology that's enough to have an even cell decomposition. Okay, so Steve Wilson proved this amazing theorem that the classifying space for complex co-bordism is an even space. So the fact that the first condition that it's homotopy groups are even, that's Milner's computation of the complex co-bordism ring. This is, was, is a complete surprise. And in fact, that's true for every even suspension, positive or negative of MU. And I, the Wilson space hypothesis that I want to make in Motivic homotopy theory really involves all those suspensions. I just didn't, I just kept it simple. I kept the statement simple for these purposes. So why am I, so this also has a highly computational proof. It's, it's been tricked out so that it's short now, but it's, it's very, there isn't a known geometric reason. But there kind of ought to be. See, what, what is the, this zero space of the complex co-bordism spectrum? You're supposed to think of that as the modular space of zero dimensional, stably almost complex manifolds moving through co-bordism. So those are, so if you really go through Tom's theory, these are zero dimensional manifolds embedded in a big complex vector space. And as they move, they can transform through co-bordisms. And I said it that way to make you kind of think Hilbert scheme. I don't know a real relationship between this and the Hilbert scheme, but it's, there should be a kind of Hilbert scheme, some sort of stabilized Hilbert scheming model for this space. And the cell decomposition would just come from the torus action on the big affine space, just like it does for the Hilbert scheme of points in the plane. So I, there should be, so the topologists don't have a good enough geometric model for this to prove this theorem, but it, it would be, that would be one way, that would be something very, very useful to have. And it does feel like there ought to be, well, like I said, some kind of, some variation on the Hilbert scheme kind of model. Okay, so now I can make the, and the, I can ask about the exact analog in motivic homotopy theory. And there's some, a little bit of guessing, but I think the, the analog of those two conditions is that pi two n minus one n of x is zero for all n. And that it, it's built, it has a motivic cell decomposition out of those spheres. So I stated that in kind of a clumsy way there. But let me just say anything with an algebraic cell decomposition has satisfies part two. So P infinity, BGL, BSL, those are all examples. It's not known, the first one we don't know is the fiber of the universal second churn class. So, oh, for that, so that I didn't, this one we don't know and we don't have, and it's mostly because we don't know the cohomology of that, of that Eileen-Libbert McLean space or of KZ of two three. So that's one, this is the first one where we don't know if it's even. We do know, all right, so the Wilson space hypothesis, is that that classifying space for algebraic co-bordism is even. And I really mean that smash S2NN for any n. Now that's known to satisfy one, so that was a theorem of Morrell's and mine a long time ago and I think there's been a lot, there's been other proofs of that. Anyway, those groups of the motivic boredism spectrum were known to be zero. The other thing I want to make a comment of, I started talking about being over the complex numbers and in all the proofs, I mean I needed rho so I needed to have some roots of unity in particular you could keep track of those, probably need minus one to be a sum of squares for various reasons, but this theorem, so you could ask is this true over the real numbers or is it reasonable to guess this over the real numbers? Then you would want the theorem in topology to be compatible with complex conjugation. So that, Mike Hill and I have checked. So that's a theorem that in topology, that's true, the analog of this is true in Z2-equivariant homotopy theory. So I don't know if I think this should be true or not, I'm kind of willing to run with it, but the indication is that this would really be true. I think one would expect this to be true over Q or maybe even over Z if this, but I think, anyway, this seems like this, there shouldn't be a field of definition issue with this. Okay, so what the Wilson space hypothesis lets you do is build resolutions out of the Motivic Eilemberg-McLean space as KZN2N. So what you do is you take the, you take what topologists call the unstable Adams-Novokov resolution and the Wilson space hypothesis says you're never leaving the haven of even spaces when you do that. And then the Voivodsky or the Posnokov, the Slice Tower for any one of those spaces is a product of these, or the associated grade is a product of these KZN2N. So you get a resolution in terms of specific Motivic complexes. In ordinary homotopy theory, that's not such a big deal, Abelian groups can be resolved into free Abelian groups at not much cost, but in Motivic homotopy theory, the homotopy sheaves can be any strictly A1 invariant sheaf and it's, they can't always be resolved in this way. So that's the Wilson hypothesis I said, it lets you make a resolution that you wouldn't know how to make. And so I've been kicking this thing around for a while. And a couple years ago, or about two years ago, Igor Krish asked me about this sort of, these compliments of the discriminant variety, whether finitely generated projective modules over those are free. And if you run that through the Wilson space hypothesis, you find the prediction is that they are. And then Igor was able to find a pretty elementary algebraic proof of that. But this was, you know, this isn't like, yeah, it predicted the red shift of mercury or anything, but it predicted something, this was sort of a reality check for it. So the point of the Wilson space hypothesis is, so I put this theorem in quotes just because we haven't written down the proof and it involves a lot of, it involves bringing a lot of sort of apparatus of homotopy theory into Motivic homotopy theory that we've sort of, we understand how it should go, but I just put it in quotes because we haven't really written the paper. We haven't written it all down. But if the Wilson space hypothesis holds, then there's no difference between Motivic vector bundles on projective space and topological ones. And that would actually be true, not just for projective space, but for anything with an algebraic cell decomposition. So Grosmonians, blow ups along linear things. I mean, all that sort of class of varieties with algebraic cell decompositions. There'd be no difference between Motivic and topological vector bundles. So that would at least give you lots of vector bundles over these Juannoli devices and not, but then I don't know about the actual projective ones. So I think I've hit my time limit, but I'll just leave you with this diagram about this problem. So, you know, the Wilson space hypothesis implies this arrow is a bijection and we can lift a lot of things to it. That gives you vector bundles on the Juannoli device. And then the question about vector bundles on projective space becomes this possibly difficult problem of descent. All right, that's it. Thanks. Are there any questions? Would you come back to the slide with the Wilson space hypothesis? Yeah, that one, yeah. That was the question. Okay. So. These are analogies. Yeah, there's also a question which I asked many years ago about kind of replacement of journal in a normal form of matrices. You can see that square matrices, an actual pjth by conjugation. Yeah. And the question is, can one make a collection if you count how many orbits on a finite field, for example, get some of q to length of partitions. Yeah. In addition, can one make a collection of affine subspaces so that any orbit intercept exactly wants this collection. And the answer is, yes, it's pretty non-trivial and it's somehow not similar to this. It's similar. Yeah, okay. You get a bunch of affine spaces. I see. Yeah, oh, that's, is that in a paper then? Somebody pulled it here a few years ago. Okay. I'll ask you later if you can help me find that. That sounds interesting. Yeah, it does smell similar. Thank you.