 that Leonardo and I started at ISOM last semester. Thanks to all of you for coming. Yeah, so just like last semester, we will have a five minute break in the middle and we will make sure to make it at least five minutes this semester. So you can actually count on getting up and get back in time to not miss the talk. Yeah, and I guess that's about it. Today, we're happy to introduce the first speaker of the semester, Ravi Vakil, speaking about bot periodicity, algebra geometrically. Next week, we will have Carl Leon speaking about Tevelev degrees. Two weeks. Two weeks, yeah. We have talks every other weeks, approximately. Thanks, Leonardo. Yeah, so, and a special encouragement from Ravi. He likes questions, but anyway, please go ahead, Ravi. Thanks. So thank you very much. Thanks for having this seminar and for inviting me. And also maybe this is a late appreciation for ISOM where I, because it's harder for me to travel, I really appreciated having this that great course semester where I could take part fully without traveling or reasonably fully. So thank you, Anders and Leonardo, for this as well. Because I am traveling on Mondays usually, I normally will miss a number of these seminars at least this fall. So hopefully I'll be able to catch some if it continues. So great. So I have an hour and so I'd like to tell you about some work separately with Hannah Larson and then Jim Bryan. And this, I really am enjoying this because this work because it's in the part of mathematics I like, which is where this seminar is located as well, which is where it's kind of the crossroads of several fields where different subjects all converge. And I know about some of them and I don't know about others. And this allows me to see a lot of things I would not otherwise see and talk to people would not otherwise talk to. So, and the tricky thing I think is I will want to pull in things that any member in the audience might not necessarily have seen. So please do stop me because I do want, I think this is actually quite, I think the subject is comprehensible so long as you're willing to hear a few words you may not have known before. Okay, so let me go right into it. So what is Bop here, I know we're not topologists here, most of us, and so I'll say just enough to get it across. It's one of the fundamental results in topology and K theory, one of the foundational results in K theory. And it's so fundamental that it looks like it comes from algebraic geometry. That should be some algebra geometric result. Jim is smiling now because he feels like I'm, I don't mean this that algebraic geometry rules everything, although that may be the case. But just that the statement is not topological sort of even more, it's even more blunt and less sophisticated than the topological. So you may have heard of it in the following context which is not algebra geometric. So for example, if you have the infinite orthogonal group and it doesn't really matter what that is, then you can write down its homotopy groups and they repeat mod eight every eight. And I wrote down the first eight and then it repeats after that. And okay, so that's something infinite, but if you have a 100 orthogonal group in a hundred dimensional space and that approximates old infinity. And so it repeats mod eight for a while and then it breaks off, it doesn't continue. So Bob Pirates, he says that in one of its many meanings is that these groups when they're finite behave nicely, but they get more nicely behaved as their groups get big and then there's this furthermore, there's this repeating behavior. Okay, but more fundamentally it's the following thing which is gonna be more relevant for us but I will explain it soon but I wanted to show you these diagrams. So more fundamentally it's about spaces and I won't say exactly what spaces are yet or what we're gonna talk about yet but we have the infinite unitary group U and again, don't worry about the details of this right yet. And so we, and when we take, there's a space called U, the space called BU and there are various spaces here oh, is there a orthogonal group and SP is symplectic. And when you take what are called base loops and a base loop is a map into our space by a circle with a chosen point and the chosen points asked to go to a chosen point here they kind of didn't agree. So we have a space of such things and the space of loops into U is supposed to be BU and the space of loops into BU is supposed to be U and when I say supposed to be, I mean is is and when I say is, I mean homotopic to topologically they are the same things. I don't really mean is but much of mathematics is about figuring out what you mean by the word is and so what they call a complex plot periodicity is this thing here, it's a two-fold periodicity loops one of U is BU, loops one of BU is U and then real plot periodicity is this eight-fold thing. I actually have no idea why one is called complex one is called real. I know people, someone knows in the audience, don't tell me tell me later, it's not gonna matter just yet. Okay, so my plan for this hour and I'm hoping for the break to take place somewhere between somewhere between these thirds is to tell you first about the two-fold periodicity complex plot periodicity and to try to convince you that once you know what the statement is you're accidentally gonna prove it. It just is, it's gonna be, whoops, it's true and then you're happy and then to make that actually true you have to define the words in your proof and so you have to sort of so the definitions come after the proof and then finally in the final third that I hope to get to and spend as much time on but it's getting richer and richer as we get into it. I want to really get into like the richness of real plot periodicity and then the division so the first two are with Hannah and the last is with Jim. But please do stop me with questions because I want the words to make sense and when I use fancy words fancy symbols that you haven't seen they're not so fancy I'm happy to tell you what it is. Okay, great. So I see there are comments but I'm gonna ignore them so I don't know if they'll tell me. And they should be across as you perhaps somewhere. Jim's gonna say perhaps, ah, great. So great. So my first goal is to take you is to figure out what this means and make sense of plot periodicity where we're gonna take loops one from you and to be you. And let me just start to get us into something I feel happier. I don't like the unitary group because I don't understand what it is but over the complex numbers the GLN deformed retracts on a new event and this is something I knew in first year of college and I forgot ever since, which is that you can so I'm just pleased to remember that you can and I hope I get this right. You can factor a matrix, any invertible matrix and a unitary matrix and an upper triangular matrix and that's using the Gram-Schmidt process and that then you can homotope your upper triangular matrix to be the identity. And so that means you can deformation retract the general linear matrices into the unitary group. So basically what that means is I can replace U of N everywhere by GLN. And now suddenly I'm in a happier algebraic world that I can understand. So what do I mean by GL when I didn't say GL of anything? Well, it's gonna be a GL infinity in some sense. So GL1 sits in GL2 sits in GL3 and so forth in the quote obvious way. So for example, two by two invertible matrices sit inside three by three invertible matrices by just nestling them in and putting zeros and one there. And so GL is technically it's a co-limited GLNs but if you want to think of elements of GL then you should think of them as being things that are invertible finite matrices and just fill them out with infinite, with the rest would be like some infinite matrix. So it's not really scary at all. Or if you want, think finite matrix but just be agnostic about the size. This looks like a two by two matrix but I'm allowed to think of it as a 10 by 10 matrix just not as a one by one. So that's what GL is. It's nothing scary. It's the way we normally tame infinity. And so what's be GLN? So this is one where the algebraic geometry in the audience would be happy and the non other people may be less happy which is it's the modularized space in some sense of vector bundles. I should say this slightly differently but what I really mean is it's a space that parameterized that tells us what the vector bundles are in any other space. So what are the rank and vector bundles on a space X? Well, the rank and vector bundles on X are exactly the maps from X to be GM. So that's what that space is. And so as an example, if I have a map point to be GLN what's a vector bundle on a point that just a vector space? And let me say that vector space. Let's say it's U. U is gonna, for example, this might be this is some dimension in vector space. And now what if I had a, if I map X to be GLN and let's say it factored through that point map. Well, that means which rank and vector bundle is it's the one coming from here which is it's the pulled back. It's a trivial vector bundle with fiber U. So that's what be GLN is. And then if you ask me a little more what I mean by space because I'm being careful about carefully not saying what I mean by space I would zoom in and say, be GLN is an arms stack. But don't let that, nowhere will that be wrong. Nowhere should that slow you down. I would say nowhere will that be wrong. But if you think that should cause you concern just stop and ask me and I will and tell me anything, tell me your worries and I will try to put your worries to us. But it's just a, it's a space. It has a top launch. If you're over the complex numbers it can have a homotopy type, if you like that. If you're over a field it can have a chow ring. It's a very, don't worry about the fact that it's not a, in the same way that an algebraic geometry dealing with varieties will tell some differential geometry don't worry about what a variety is. It's you basically what it is. It's the same thing that's happening. Okay. So maybe I'll stop and ask if there's anything stack-wise, anything, maybe I shouldn't stop but tell me if there's anything you feel nervous about this so far. Okay. So let me just get some practice with the concept. So here's a map from BGLM to BGLM plus one, which is going, so I'm going to take the map corresponding to if I have a map from X to BGLM I need to give you a recipe. If I give you a vector bundle of rank N I'm going to give you a recipe for a vector bundle of rank N plus one. And my recipe is going to be add the trivial point. So that recipe is the data of this map. So this map is add trivial I've defined this map to be add trivial bundle of rank N. Okay. Now we're ready for BOP periodicity. So BOP periodicity says loops to a BGL. I mean, what BGL is now it's a space of these vector bundles should give me a, I want to say that is going to be the same as BGL and there's a Z here which Jim pointed out in the comments and maybe let me not worry about that and I can explain what that Z is later. But let me explain more importantly now that we know what BGL is it's basically vector bundles. What is this loops to? Well, when I see loops to and I put myself in the mind of a topologist loops to is a base means it's a map from a sphere with a chosen point. So it's a, it's a, it's a base map from a sphere. So it's a sphere with a mark point mapping to BGL. But when I see a sphere with a mark point that looks to me like Cp1. And so, and if I want to be sophisticated and think algeo geometrically and not be hung up on what field I'm thinking about I'll just call it P1. So what I'm going to say is that I want to just consider maps of P1 to BGL with a mark point infinity. And that's what it's going to be. And if you want me to go back to topology land I just say no problem we work over the complex numbers we specialize in complex numbers. And so now what I'm saying is this is maps of P1 with a mark point P infinity mapping to this thing. And now I made this into an algebra now this is an algebra geometric statement. I have a, I know what this object is I know what this object is I just need to figure out what the map is and show it's a nice morphism. And for experts who know about these things with the way I've defined it this is the affine grass money. So it's a sign that, you know we're on the right track that we've we're discussing something fundamental. Okay. So our goal is to define this map beta this bot map beta and show it's a nice morphism. So in other words, the bot map beta was supposed to be a map from loops to a BGL to BGL. And what that means to show you what the map to figure with the, to define the map it means that any time I give you a map to loops to a BGL and I'll translate this in English in a second I get a map to BGL. So the translation and okay maybe this isn't English but it's as into it's out of topology language. If, so if, if I give you a vector bundle on and I'm agnostic let us write I give you a vector bundle on P1 cross X so we've got X and I give you a vector bundle on P1 cross X. You've got to give me a recipe to give me a bundle on X. That's what I'm asking. So, and so let me say that again you give me, I want a recipe. So the recipe says, if I give you an X and P1 cross X and a vector bundle on P1 cross X give me a bundle on X. Well, how would you do it? And maybe I feel like I want to ask if there's an idea like there's no I don't want the perfect idea I just want a reasonable guess as to how you would get a bundle on P1 cross X and turn it into a bundle on X. Just use the mark point, right? Ah, that's a great, you could pull it back. So great. So one way of doing it is you would use a mark point and so there's your section. Let me call it section S and you could pull back this bundle. And now if we want it to be trivialized this should be a trivial bundle. So one way to do it is by pulling back the mark point we should get the trivial bundle. So now I want a different way to get a different bundle potential. Actually, I should say that that was just a pi naught versus pi two. So if I give you a bundle on P1, right? So, and I feel like if I asked the right leading question I shouldn't ask that. No, I think that was already a good idea. Any other good ideas like that? Would it be possible to remove the section and then take something which is like seeing variant and descend? So great. So Nicholas saying remove the section. So the intuition is to remove the section and he somehow was saying remove the part that was trivial and see what's left. And I want to extract from that something like a derived category kind of idea what he's trying to, what he's doing great. So we have a part of this bundle is going to be the base point. And then we have the rest of it is going to be we want to sort of do some drive category thing where remove the trivial one. Okay, that's a good idea. That's and so that idea is in the right sense correct as well. So I don't know if it's worth taking even I can think of at least two other correct answers that are global that sound different. So yeah, so that answer will be correct as you will see in a second. Maybe you don't stop yet if anyone wants a different has a different idea. Let me say it. Okay, let me just say one maybe I'm more naive like I would have thought so if I were so my naive idea would be push it forward if the bundles generate a global section you just push it forward and you get a vector bundle downstairs. But if I were a fancy, I would say something would this be fancier than Nicholas answer? I would say yeah. So the next little fan says beyond Nicholas answer would be some sort of drive category push forward or some sort of flourish rate. And so if you put those four answers together all four are sort of correct. You take Nicholas answer or subtracting out I didn't catch you, we said the first answer and you subtract out the trivial and you're taking the push forward minus the trivial is going to be exactly what it is. So let me tell you the answer in the first case and then you can translate it to the others. So you have a vector bundle on P1. One thing you may know about vector bundles on P1 is that they are direct sums of line bundles of AIs. And so maybe, okay, if it's non-negative by which I mean if each of these guys is something to the summands is non-negative or generally global sections in different ways of saying it, one way of getting a bundle and this is unlike the answers that you suggested this example does not this idea just doesn't work all the time. But if it happens to be non-negative on a big open set words non-negative then we get a bundle and that would and so that's great and I give you a bundle. And so there's a problem with what I'm saying actually, yeah, there's a problem with the three answers so far all of which go away when you think about it which is we are agnostic about the rank of the bundle. So what if instead of E we added a trivial bundle because those are supposed to count we're supposed to be agnostic about we're supposed to be able to add trivial bundles and that counts as the same bundle and all in this answer when you push it forward E plus O you push forward E and the trivial bundle and you get the push forward to be in the trivial bundle and so when you adjust it in a trivial way that wasn't supposed to count the output you get basically the same bundle and if you did the same thing with the if you pay attention to the two suggestions they also play well with respect to adding trivial bundles. So we get a well-defined bundle we had a well-defined push forward now what sucks with my idea but not with the other ones was what if you was not positive and couldn't just push it forward but what you do if you have a bundle that's not positive and you wish it were you just twist it up to make it positive. So you'd instead say twist up by K and make it positive and so then you'd be happy maybe you'd be unhappy because you always have to twist up infinitely much to make every bundle work but you could instead all I need to check is that when I twist it up I get the same answer. In other words, all I really need to check is that if I push forward E of minus one and you have K minus one I get the same as pushing forward E of K. So if that were true, well it's not true then we would get the same push forward. However, when we push it let me just write this down we take a short exact sequence you have the bundle twisted by minus one we have the bundle and we when we push forward and actually here we have the very first that is this is the very first suggestion that we got is how to get a bundle on the base it was pulled back the section from infinity and now that's the trivial bundle. So the bundle we get by pushing forward E or E of minus one are the same bundle just they're extended by a trivial bundle it's not a direct sum but it's essentially direct sum and you get like there's an extension class you can send to zero so up to homotopy basically the same thing. So okay, so what that means is let me just go back a second we now have a well-defined map from this that's our beta. I said it in terms of just take your family twist it up and push it forward. Nicholas said it by remove the infinity section figure of what's going on away from it or you could also say it by take the lower shriek and in K theory. So basically all the answers you try to come up with will give you the right thing and then the magic is that this map you now have to find a map from loops to a VGL to VGL. And once Bob tells us there should be such a map we read no choice but to make it. And the magic is that this is the reversible construction and this is something which is I feel like it's a surprise and it's amazing insight and one of the again the many things about the way mathematics works is so Leonardo gave the reference to the paper which explained how to think about this the right way. I don't think it's possible he will have no recollection. The question was not appreciably it would not have sounded like this question at all perhaps he has no idea what I'm talking about right now perhaps. So, but I wanna explain this idea that came from the paper that he said to look he told us to look at. And it's gonna be three pages and there's like a lot of insight here so I hope you stick with me but if you don't you can come back and make pages. Okay, so I wanna tell you how to describe a modularized space of vector bundles on P1. Oh, I should say the name present here is stroma. This is distilled I don't know what the stroma would recognize what I'm about to say but and I also feel like this is so fundamental that I'm sure people knew it before stroma but I don't know where it was said and in what way but so okay, so consider vector bundles on P1 and I want the vector bundles to be rank n fix the rank fix the degree just for concreteness and let's make them non-negative in a way that I said so the summands are all positive so E is a bunch of positive things and that means by Raymond Rock that we know how many sections E has and E twisted by minus one so and so let me just call this vector space A we have a vector space of dimension DN and a vector space of dimension D plus one N which are just my global sections of my vector bundle and so I just wanna give that one a name and again, we've required that it be trivialized infinity you have this base point requiring that this vector bundle over P1 over infinity is this fixed bundle U so of rank n, we have a rank n thing so so far I've given you the data of a dense open subset of our space of bundle rank n bundles on P1 of degree D open because I required that they be positive that they be that all the summands be non-negative and I'm gonna add one more bit of information this is something Jim calls section friendly and so I also further this is generated by global sections there's sections of our vector bundle which give me all my give me there's section of the vector bundle which give me all the elements of my vector space over infinity and I'm just gonna choose a splitting or section of that map so I'm gonna I just wanna map from you to my space of sections and I just wanna point out that because this is a surjective map this is not a serious addition I'm making all I have a surjection of vector spaces then asking for a mapping of the direction is like an affine bundle so if I'm doing topology it's a choosing element of your vector space and choice is topologically contractable so I'll add that one extra bit of information and now I come to something which I think of as the sort of coming up again and again in math which is anytime something is complicated like a modular space you want it to turn into linear algebra and so I claim that I can recover I will name stroma claims that I can recover my vector bubble from linear algebra data from the following quiver or linear algebra description so I claim I can recover from the following thing I've got a vector space and look at the part that's in red I've got a vector space of dimension DN and I've got a fixed vector space of dimension N so pick for me so choose any endomorphism of A and any map from A to U and so now I'm gonna tell you what E is I'm gonna build E by taking the trivial bundle on P1 A underline and the trivial bundle on P1 U underline and I'll map A to A plus U by okay what's this map gonna be? It's like it's degree one so I need to like put coordinates on P1 so that part's identity on A did I screw that up? No, I didn't I feel like I screwed it on No, no, that's right, good so it's the map to the A is gonna be the identity times X dot minus alpha times X one I tell you the map from this part to this part my map from here to U is just J times X one so again the magic is that this works and I'm not, so we have an open condition that this should be that the co-kernel of whatever this map is should be a vector bubble that's an open addition and the magic is this recovers E and to prove it, it's just a calculation you just check, look it recovers E, it just worked we already know what A was we already know what U was I can tell you what J and alpha was the actual proof is a triviality once you know it's true but I have no idea in my gut how I would have invented it so let me say it again the data of a vector bump of this sort is nothing more than a vector space A a vector space U, these two maps and that's it with some open condition and how do I recover E? It's a co-kernel of this map how do I reverse it? Well, you already know what it is well, I can tell you how to reverse it so the translation again of what this is my third and final page about this trick is that the space of such vector bubbles is just the space of matrices it's a linear space, affine space except it's an open subset through some explicit condition module O that JL Translation two is equivalently the space we're looking at the space of bundles that are non-negative is an affine bundle the open subset of an affine bundle would be JL and translation three which is the way stroma would say it which is that you take your bundle it's generated by global sections this is for the algebraic geometry it generated by global sections so it turns to so that means that there's just rejection like this what's the kernel? The kernel is you push forward you have minus one pull back it's just about minus one and that's what it is so it looks different but all these three things are different ways of saying the same thing but the upshot is that as you twist and twist and twist you get bigger and bigger opens and affine bundles over BGL and the result is that the result is that is that beta is nice and nice beta gives you a nice amount of space so just to sum up what happened or all I want you to remember what happened is we figured out what the map was cheaper we just guessed what it had to be one of the two directions this is the map that Bob came up with in history, Atea came up with the map and the opposite direction in this one, what we have to do is we just rely on this bit of magic which comes up repeatedly later on okay, so that is so now we are at about 1.30 or 4.30 so let me before calling for a break let me just say that I want to say that that concludes a proof of block periodicity with the one caveat that what I said doesn't necessarily make sense because none of the words I didn't say what any of the words meant so what is about to happen after the break is I will say what I meant and then the words I make up are designed to make the proof just work without any change and then I can declare a victory and it's not cheating because I actually as a consequence get all these statements that we want to actually take place so you basically say here's what the properties I need to have true make the definitions work and look every part of the argument makes sense there so that's what's going to happen after we have some questions and apply on a break is that a good time on yours for yes this is excellent yes very nice yeah before we have the break any quick questions to Ravi I don't see anything in the chat and I don't hear anything but yeah it was very nice and clear all right let's see