 Okay, so I think everybody is back. So now, so again, do not hesitate to ask questions. Just I wanted to say that if you use the Q and A Chan, then you can put anonymous questions for those who are shy, but you can also use the public Chan if you want. So now it's time for Tina Gheart from the University of Michigan and she will speak about the algebraic K theory and trace methods. Thank you. Thank you all for joining me today. So I'm going to be giving three lectures over the next few days. And my goal for this lecture series is to introduce the algebraic K theory and the so-called trace method approach to studying K theory. And I'm going to talk about some exciting recent developments in these areas. These talks are really aimed at early career researchers like graduate students and postdocs. So I'm not going to assume any familiarity with algebraic K theory, and I really want to start right at the beginning of the story. So what we're going to start with in the lecture today is to address the question first, what is algebraic K theory? So I'm going to talk about the algebraic K theory of rings and I'm going to let A be a ring. So I'd like to take a historical perspective for a while and talk about the development of this field. Throughout time. So we're going to start really at the beginning of the story of algebraic K theory. So we're interested in studying rings. And when you want to understand a ring, one thing you could consider about it is modules over that ring. So I'm going to let P of A denote the monoid of isomorphism classes of finitely generated projective A modules. So recall that a projective module is a direct sum end of a free module. And I claim that this is, excuse me, I claim that this is a monoid, which means we have some sum operation. So if you take the direct sum of two projective modules, you get another projective module, but this doesn't have inverses in it. So it's a monoid, but not a group. So what is a zero? Any question? Do you assume that your rings are associative, commutative, et cetera? Associative, yes. Commutative, no. So what's the zero of algebraic K theory of our ring? Well, it goes back to work of growth beak. And the definition is as follows. The zero of algebraic K theory of a ring A is the group completion of this monoid. So we have this monoid and group completing essentially means, well, I didn't have inverses in it, but I can just formally add those in to get a group. So let me illustrate that perhaps best with an example. So let's say our ring was actually a field. So for a field, F, what happens? Well, first I wanna consider this monoid of isomorphism classes of finitely generated projective modules over a field. Well, modules over a field are vector spaces and up to isomorphism, they're classified by their rank. So P of F is just isomorphic to the natural numbers. And so what is the zero of algebraic K theory of a field? Well, I'm supposed to take the natural numbers, which doesn't have additive inverses and formally add them in. And what do you get then? Well, you get the integers. So the zero of algebraic K theory group of any field is the integers. So the definition of, so that's zero of algebraic K theory and the next sort of group to come along in history is the first algebraic K theory group. So this comes out of work of Whitehead and Bass in the 1950s. And here's what the definition of K one is. So the first algebraic K theory group of a ring is what you get when you take GLA, the infinite general linear group and abelianize it. Now it turns out that the abelianization of GLA can also be written as GLA modulo subgroup EA generated by certain elementary matrices. Okay, so this is the first algebraic K theory group of a ring. So in our historical overview, we're sort of in the 50s and in the 50s and into the 1960s, people started to see that these algebraic K groups, although they're defined purely algebraically, started to have interesting applications to topology. So one illustration of that is through the famous S-cabortism theorem. So the S-cabortism theorem was proven independently by Mazer, Stallings and Bardin in the 1960s. So what question is the S-cabortism theorem aiming to answer? Well, it's the following. Let's say we're in dimension at least five. So for n greater than or equal to five. If w is a, let's say, a compact n plus one dimensional cabortism between two n-manifolds, sorry, let me try to spell, between two n-manifolds, which I'll call x and y, such that the following is true. When you look at the inclusion of x into the cabortism and the inclusion of y into the cabortism, that those are both homotopy equivalences. So that's set up where you have a cabortism with that condition that those inclusions are homotopy equivalences. That's called an H-cabortism. And what the S-cabortism theorem is addressing is the following question. When is the cabortism trivial? So what do I mean by trivial here? Well, the question is asking, when is the cabortism really just a cylinder on x? Okay, so that's the question of the S-cabortism theorem. And it turns out that it's not always the case. There's an obstruction to that being the case. And the obstruction lies in a quotient of algebraic k-theory. In particular, it lies in a quotient of the first algebraic k-theory group of the group ring on the fundamental group of the manifold x. So this is one of the first instances we see of this purely algebraic construction of algebraic k-theory having really interesting applications to sort of geometric topology. So there was a lot of interest in these groups in part because of these applications. And around that same time in the 60s, Milner gave an algebraic definition of the second algebraic k-group, k-2 of A, in 1967. In the interest of time, I'm not gonna go into what Milner's definition of k-2 is, but the thing that you should know about it is that it's again a purely algebraic definition in this case in terms of generators and relations for the second algebraic k-group. So one question you might have in your mind looking at this is, well, I've told you that there are these groups k-0, k-1, and k-2. The naming certainly suggests that they're related to one another, but why are we calling all of these things algebraic k-theory? So one answer to that is that at the time, we're sort of in the 60s in history, at the time there were known relationships between these groups. And there were a lot of them, but I wanna mention one of them. So there were a lot of known relationships between these lower k-groups, and here's an example. Let's say we have A, a ring, and I, an ideal of A. You can algebraically define what are called relative groups. So there's a way to algebraically define groups k-0 of A, rel i, k-1 of A, rel i, and k-2 of A, rel i. And then it was known that there's a long exact sequence of the following form. So there's a long exact sequence that relates this relative algebraic k-group, k-2, to the k-theory of the ring A, to the second k-theory of the quotient, A mod i, and then it continues on to the first algebraic k-theory, the relative group, the first algebraic k-group of A, and so on. Okay, so this is the kind of thing that suggests that these groups are all instances of some common object, which we're calling algebraic k-theory. So in the 1960s and 1970s, of course, the big question became, well, we have k-0, k-1, and k-2, and it turns out they've been really useful for applications to things like topology. And so people were asking, well, can we define all algebraic k-groups? So the question was, can we define k-n of A for all n greater than or equal to zero? Now, what would we want out of this? Well, we'd want it to agree with the known definitions. So we want this to be agreeing with the known definitions of the lower k-groups, k-0, k-1, and k-2. But more than that, I needed to also extend all these relationships that we know between the k-groups. So, and extending the relationships, between them. Okay, so that was the question. Now, looking at these properties, for instance, the one I've written down, I haven't told you some of these other relationships between them, but looking at these properties, they start to feel reminiscent of the kind of thing you see in topology. I mean, for instance, this long exact sequence, it's the kind of thing we typically see in a topological setting. So that's not a coincidence. So the first definition of algebraic k-theory is due to Quillen. And Quillen did this work in the early 70s. And here's Quillen's definition of higher algebraic k-theory. Quillen says the nth algebraic k-theory group of a ring A should be the nth homotopy group of BGLA plus when n is greater than zero. So let me unpack that a little bit. What is this? Well, GLA is the infinite general linear group again. We saw that earlier. BGLA is its classifying space. So we have this method in topology of associating a space to a group, and that's the classifying space. The superscript plus is something that we now call Quillen's plus construction. And what that is, is it's something that you can do to a space that doesn't change any of its homology, but does something very specific to the fundamental group. So it kills a maximal perfect subgroup of pi one. Okay, so this is the first definition of higher algebraic k-theory. We often call this now the plus construction, definition of k-theory. Now, if you go into the literature though, if you're interested in algebraic k-theory and you look into the literature, you'll see that these days there are lots of definitions of algebraic k-theory, lots of different perspectives on this. And that's in part because now we can take the algebraic k-theory of a wide variety of things. So I'm talking right now about the algebraic k-theory of rings, but these days there's a notion of algebraic k-theory for spaces, for ring spectra, for varieties and schemes, for exact categories, really for any category with a notion of co-fibration and weak equivalence, et cetera. So we can take algebraic k-theory of a lot of different kinds of objects now. And partly for that reason, there are lots of different ways of defining algebraic k-theory. But for any of those definitions, if you restrict your attention to rings, it still agrees with Quillen's original plus construction definition. So this is a perfectly fine way to think about the algebraic k-theory of rings. So Quillen made this definition of algebraic k-theory and at the same time, he did a really beautiful calculation, which is the following. Quillen computed the algebraic k-theory for all finite fields. So he computed that the algebraic k-theory for the finite field with q elements is, well, we've already seen that it's the integers in degrees zero, but he computed that it's z mod q to the i minus one for n equal to i minus one. And it's zero when n is greater than zero and even. And I should note that Quillen later won the fields medal for his work in algebraic k-theory. Okay, so now we're in the early mid-70s in our historical overview. And we're feeling really good about things because now at last, there's this definition of higher algebraic k-theory. And frankly, it looks simple enough, right? This was supposed to take home Joby groups of some space. And so then of course, people try to start computing it. We have Quillen's beautiful calculation for finite fields and we'd like to compute the algebraic k-theory of other rates. So what else might we ask for the k-theory of? Well, Quillen tells us the k-theory of z mod p, but maybe we should look then at the k-theory of z mod p to the k. There's a funny story about this that one of the senior members of my field told me, which is that he was advising PhD students in the mid-70s and he gave this to one of his PhD students as a thesis problem. Compute, extend Quillen's work, compute the algebraic k-theory of z mod p to the k. The PhD students struggled, had a really difficult time with it and ultimately just really got stuck. And this is not a sad story. The student moved on to a different project and eventually graduated. This happens, of course. And I think at the time the advisor also realized that maybe he didn't know how to solve this problem with algebraic k-theory. And it turns out that now in retrospect, almost 50 years later, we know that this was a really bad thesis problem for a PhD student in the 1970s because still today, nobody knows how to do this. So this is still unknown. Okay, so maybe that was too ambitious. Maybe we should just start with the k-theory of z mod p squared. Some things are known about that. It's known in low degrees and there's some stuff known about it, but in some broad sense, this is also not known. Now, what's another brain that we might really be interested in the k-theory of? Well, we might be interested in the k-theory, of course, of the integers. Now, this is the subject of an enormous body of work that has been going on for 50 years and a lot is known about this now, but this is still not completely known. And I will come back and say more about that in a few minutes. So if this were not my field and if I was in some other field and I was looking at this, I mean, something that I might think to myself is, you know, so why are we trying to compute this? I mean, just because we can define a ring invariant doesn't necessarily mean that it's a good idea to study it. And I've just argued for you that algebraic k-theory is really difficult to understand in even these like really pretty simple situations. And so maybe this just isn't a useful invariant of rings. Well, so why do people study algebraic k-theory? I mean, one answer is, of course, that the mathematics around it is beautiful. But maybe even larger than that, algebraic k-theory is one of these sort of magical objects in mathematics that appears across mathematical fields. So k-theory, although, you know, we've seen it in the variant of rings, it has applications to many areas in mathematics, including homotopy theory, number theory, algebraic geometry, geometric topology. And so part of the interesting k-theory and k-theory calculations is because of those connections. So. So there's a question. He may be, can you say a word about the tools that Quillen used to compute the k-theory of finite fields? Yeah, so, right, it's a very interesting story. Quillen sort of, well, I don't know. I wasn't even born then. So maybe I shouldn't claim what he was thinking, but his work on the k-theory of finite fields came up in work that he was doing on another conjecture, having to do with the image of what's called the j-homomorphism. And it turns out that he was able to identify this BGLFQ plus as a kernel of some atoms operations related to topological k-theory. So part of the reason that it's difficult to sort of extend Quillen's work to other situations is because it was sort of specific to the situation of these finite fields. So it's beautiful work, and by what he does is essentially he's able to identify this space, BGLFQ plus as the kernel of something having to do with atoms operations, and then that's much easier calculationally to get your hands on. So that's sort of the idea, but also that doesn't extend broadly to other k-theory calculations. Right, so what I was saying is I was claiming that algebraic k-theory has these connections to other fields of mathematics, and I'd like to tell you a couple stories about what I mean when I say that it has connections to other fields. So let me give you an example, which comes from number theory. So this example is a conjecture, and it's called van der Wer's conjecture. So it's called van der Wer's conjecture, but actually it's due to Coomer in 1849, and he wrote this conjecture in a letter to Kroniker, and here's the conjecture. It says for a prime P, let's let k be the maximal real subfield of the rational adjoin a primitive P through to unity. Then the conjecture is that P does not divide the class number of k. So I'm not gonna dwell on what any of that means is sort of not necessarily important for what I'm saying today, but what I wanna say about this conjecture is that this is a very important conjecture in algebraic number theory. So this is the kind of conjecture that people prove theorems based on, assuming van der Wer's conjecture is true, then blah, blah, blah, blah. So this would be a huge step to be able to establish van der Wer's conjecture. It is thought that if there are counter examples, they are indeed rare. So it has been verified computationally for all primes less than 2,147,483,648, which is not just some random number, that's two to the 31st. So if there are any counter examples, the primes would have to be bigger than that. There is, I should know that there was a couple months ago acclaimed proof of van der Wer's conjecture on the archive. I haven't had a chance to look at that too much myself and since it's not published, let's consider van der Wer's conjecture to still be open. So this has been a longstanding conjecture in algebraic number theory. So why do I mention this in the context of what I'm talking about today? Well, Curahara proved in 1992 that van der Wer's conjecture is equivalent to another conjecture. So Curahara proved that van der Wer's conjecture is equivalent to another conjecture. And that conjecture is that the K4M, the algebraic K theory groups of Z of the form K4M are all zero for M greater than zero. This to me is fascinating. I mean, if you think about that algebraic K theory of the integers, it was defined as some homotopy groups of some space. And when you look at van der Wer's conjecture here, you don't see any homotopy theory. I mean, you don't even see the integers, right? So this is sort of a fascinating connection between algebraic K theory and a deep conjecture in algebraic number theory. And it's telling us that the K theory of the integers is somehow capturing some very deep arithmetic information. Okay, so that's one kind of connection that I mean when I say algebraic K theory appears in unexpected places. Let me also mention connections between algebraic K theory and algebraic geometry. So one of the subjects of this summer school is motivic homotopy theory. And the connection between algebraic K theory, how it appears in algebraic geometry is related to motivic homotopy theory. So for those who aren't motivic homotopy theorists in the audience, what is broadly speaking the idea of motivic homotopy theory? Well, it was an effort originally due to Morel and Gwaiwokski to bring tools from topology into the study of objects in algebraic geometry. So I'm gonna make a very rough dictionary of what that might look like. So I'm gonna make a little dictionary here of topology and algebraic geometry. And what am I gonna put in my dictionary? So let's think about our first course in algebraic topology. And what do we study? Well, we're interested in studying invariance of spaces. And what's usually the first invariant of spaces that you learn? Well, it's usually singular co-homology. Now, when you go on in algebraic topology, you learn, well, they're actually generalized co-homology theories. And so maybe we also learn about the invariant topological K theory, which has to do with vector bundles on our space. Now, if we wanna translate these kinds of tools into algebraic geometry, what are the analogs going to be? Well, maybe in topology, I said my fundamental objects were spaces, but in algebraic geometry, we might wanna consider varieties. And whereas we had singular co-homology, in algebraic geometry, we have a co-homology theory that we can use to study varieties called motivic co-homology. And then the claim that I wanna make is that the analog of topological K theory in topology, in algebraic geometry, is actually algebraic K theory. So what is the content of that statement? Well, one way to think about what the content of that statement is, is, well, one tool we have in topology that relates singular co-homology in topological K theory is the Atiya-Herzburg spectral sequence. Herzburg, did I do that correctly? Atiya-Herzburg spectral sequence. And what is that? Well, roughly speaking, it's a spectral sequence that has E2 term in singular co-sorry, co-homology, and converges to topological K theory. So there's a large body of work on trying to establish the algebraic geometry and the log of this kind of Atiya-Herzburg spectral sequence. And it turns out there is such a spectral sequence, which is often called the motivic spectral sequence, which has E2 term in motivic co-homology and converges to algebraic K theory. So in other words, the fields of algebraic K theory in motivic co-homology really inform one another through these kinds of tools. Okay, so those are just a couple of stories of how algebraic K theory is appearing these different branches of mathematics. We also saw one through the Eskibortism theorem of how it appears in geometric topology. So hopefully I've convinced you that these algebraic K theory groups, although they're difficult to compute and challenging to understand are important and interesting to study. So what I want to move on to then is a different question, which is, well, how are these groups computed? So how are algebraic K groups computed? With that, I'm also gonna take a bit of a historical perspective and sort of both go through history and talk about the development of this method of computing that I want to focus on in this series. And I'm gonna drag an example through history with me just so you can see really concretely how at different points in time the computational powers changed. So the example that I'm gonna drag with me is the algebraic K theory of Z join X, mod X to the M. So this is truncated polynomials over the integers. Okay, so in the 1960s and 1970s, if you look at the K theory literature from that time, what you see is you see a lot of low dimensional calculations. So around that time, most of the literature is low dimensional calculations using algebraic tools. So what you see is a lot of calculations of K zero, K one, K two, even after Quillen's definition of higher algebraic K theory because it wasn't really understood how to compute with that. So for example, in this example that I'm going to carry with us, there's a theorem due to Geller and Roberts from 1979 where they compute the second algebraic K theory group of Z join X, mod X to the M, rel X and they compute that that's Z mod two. So let me just say a brief thing about this relative group. This K theory of Z mod X to join X to the M has a copy of the K theory of the integers sitting in it. And the relative K theory is basically saying what do you have beyond that? So that's what they're computing is what do we have besides a copy of the K theory of the integers. Okay, so that's a calculation of the second algebraic K group and they do that using the algebraic definition of K two. But we'd really like to understand all these higher algebraic K groups. So really our goal is for a ring A, we wanna be able to compute the algebraic K theory of A for all Q greater than or equal to zero. That's what we'd like to be able to do. And I claim that there's a tool that's gonna help us do so. So I claim that a tool that can help us and has helped over the years in these calculations. I'll try to someone raise hand. Sorry, maybe, can you ask your question to an A? All right, go on. Okay. I claim that a tool that's gonna help us do this is Equivariant Stable Homotopy Theory, which is also one of the themes of the summer school. So what is Equivariant Stable Homotopy Theory for those that are not familiar? Well, it's an effort to study spaces or spectra that have a group action on them. Now, if you think about that too hard, you may be a bit surprised because I've defined K theory already. I've talked about the question that we're trying to address and nowhere were there any group actions. This wasn't an Equivariant question. And that's a fascinating thing that we're seeing a lot recently is that sometimes Equivariant tools can be used to address questions which on the surface are not Equivariant questions. So algebraic K theory, it turns out, is gonna be a great illustration of that. But another one that you might be familiar with is Kilhopkin's and Rabinel's solution to the Curvair and Variant One problem. So that's a classical problem in differential topology that really is not an Equivariant question, but it turned out that they use Zmod 8 Equivariant Stable Home, which will be theory to solve that. So that's sort of a surprising thing. The specific approach I wanna tell you about is something called Trace Methods. And the Trace Methods approach originally comes out of ideas of Goodwillie and it's also due to Boxted, Shang, and Madsen. Okay, so the idea of the Trace Method approach is simple enough. The idea is the following. Algebraic K theory is really hard to compute directly from the definition. So maybe we can approximate it by something that's more computable. Maybe there's a question for in the preceding example, is the relative K1 trivial? Oh, I have to think. No, right, no, I just picked K2 as an example of A low-dimensional calculation. There were other low-dimensional calculations, K0 and K1 were also known at that time. Although I couldn't off the top of my head give you the citation for who did those calculations back in the 60s, but it was known. Okay, so what's the idea of Trace Methods? Well, the idea is that we wanna approximate K theory by something more computable. And we'll see that what we're gonna do is we're gonna make successive approximations to move closer and closer and closer to K theory with the idea that hopefully eventually we get close enough that we can actually recover something about the K theory itself. So what is our first approximation going to be? Well, our first approximation is gonna be via another familiar ring invariant, which is something called Hock-Shield homology. So let me tell you what the definition of Hock-Shield homology is. So I'm gonna let B sick of A denote what's called the cyclic bar construction on my ring A. So what is that? Well, it's a simplicial abelian group and it's defined as follows. So for a simplicial object, I have to tell you what are the simplices. So at the Rth level, this thing is R plus one tensor copies of A. And then a simplicial object, I have to tell you what it is at each level and also the face and degeneracy maps relating them. So what are those maps? Well, the face maps, the Ith face map takes a tensor A0 through AR. And the way to think about these face maps is most of them just take two adjacent elements and multiply them together. So the Ith face map takes AI and AI plus one and multiplies them. And that makes sense as long as I is less than R. What does that last face map do? Well, that last face map is gonna bring the last element around to the front and then multiply. So it's going to be AR A0 tensor A1 through AR minus one. Okay, so those are the face maps and we also have degeneracy maps. So the degeneracy maps just insert the unit after the Ith coordinate. So AI, tensor one, tensor AI plus one and so on. So inserting the unit and that makes sense for all I. Okay, so we have these face and degeneracy maps and it's nice and helpful maybe to have a picture in your mind of what this looks like. So what I'm saying is that my simplicial object looks like this. I've got A and then A tensor A and then A tensor A tensor A and so on. And I have these face maps in this picture. They're going downwards, downwards direction. They have a bunch of face maps and then the degeneracies go in the opposite direction. Okay, and that picture continues. Now I wanted to notice, note that this, that this thing also has a cyclic operator on it. So it has a map, TR, so on the Rth level it has a map that's gonna take my tensor and just rotate the last element around to the front. This goes to AR tensor A0 through AR minus one. Okay, this is the cyclic bar construction. Remember my goal was to tell you what is hock shield homology. So what is the hock shield homology? Well, from this cyclic bar construction I can get a chain complex. I'm gonna call it C of A. And what is it? Well, it's the cyclic bar construction. But right now I have too many maps. It's not a chain complex the way I've presented it so far. But I claim that if you take the alternating sum of these face maps that that squares to zero. That gives you a chain complex. You can check the combinatorics of that yourself if you want. And so what is hock shield homology of a ring? Well, hock shield homology of a ring is just the homology of that chain complex. Now the dolde-con correspondence tells us that we have another way of characterizing this which can also be useful for us. The dolde-con correspondence tells us that the homology of that chain complex is also the homotopy groups of the geometric realization of that simplicial object. Okay, so that's what hock shield homology is. Let me make a note about this, which is the following. So I said that this was a simplicial abelian group and it is, but it's actually more than that. So this cyclic operator that I talked about here that's not part of the structure of a simplicial object. That cyclic operator actually makes it into what's called a cyclic object. So the cyclic bar construction, hence the name, is a cyclic object. And why is that important for us? Well, it turns out that by the theory of cyclic sets due to con, that when you geometrically realize a cyclic object, it has an S1 action. And it turns out that S1 action is gonna be important for us. So this is the first time we're seeing a group action, maybe the equivalence is starting to come in, but it will be later, we'll see why that's important. But I wanted to mention it here that we have that. Okay, so the goal remember was that I was trying to approximate algebraic K theory. So the next question is, well, in what sense is this an approximation? So there's a map from algebraic K theory to hock shield homology. And that map is called the Dennis Trace. Maybe I won't go into total detail about that, but let me give you some idea of why you'd have some map like that. So the rough idea is that this Dennis Trace map is induced by a simplicial map that looks like the following. So remember algebraic K theory was something about BGLA. And I have a simplicial map from the nerve on GLNA to the cyclic bar construction on N by N matrices with entries in A. And then there's a map I claim from there to the cyclic bar construction on A. So what are these maps? Well, what is an element of this nerve look like? It looks like an R tuple, G1 through GR, in simplicial degree R. Now that thing needs to map to an R plus one tuple in the cyclic bar construction. And what it maps to is it maps to, you take the product G1, G2 through GR, the product of all of them and take the inverse, and then tensor G1, tensor through GR. And so that gives you a map from, that gives you this left-hand map. And then what is the right-hand map? Well, it's some kind of multi-trace. And that's where the trace comes into the trace methods is from this kind of trace construction. So I'm asking the question maybe a bit late, but someone asked why do you use trace as for to call this map? Yeah, right. So here it is. So this map is really a common notion of a multi-trace. So this map from N by N matrices, sorry, this map on the cyclic bar construction is induced by a map from these tensors of N by N matrices to tensors of elements in A that is sort of a classical multi-trace, where you sum over a bunch of indices and take some trace here. I'll get the formula wrong if I try to write it off the top of my head, but it's a classical sort of multi-trace. And that's where the trace comes from and the sort of trace methods. Now, of course, this is a map from GLN and we really wanted a map from the infinite general linear group. So you need to take some co-limit as N goes to infinity. And then also we need to pre-compose with a Haravich map because we wanted a map from Homotopy Group. So there's a bit more to it than what I've written, but this is what induces that Dennis trace. So the plus minus on the Dennis trace is the following. So on the one hand, we were trying to compute algebraic K theory by something that's more computable and Hock-Shield homology is much more computable than algebraic K theory. So that's good. The downside is that it's just not a very good approximation. And maybe that's really not so surprising because algebraic K theory was this invariant defined using Homotopy Groups. You know, really it was a topological invariant and Hock-Shield homology is, as we just saw, something purely in homological algebra. So we wouldn't necessarily expect Hock-Shield homology to capture all the deep information in algebraic K theory. But it turns out there is something that we can learn about algebraic K theory from even this algebraic perspective. So Goodwillie in the 1980s showed the following. So Goodwillie proved that the Dennis trace lifts through what's called negative cyclic homology. So in other words, I had this map from the algebraic K theory to the Hock-Shield homology and Goodwillie proves that actually factors through another invariant, which is called negative cyclic homology. And he further proves that rationally, negative cyclic homology is in some situations a good approximation to algebraic K theory. Okay, so let me unpack that a little bit. I have a good question here. So can you interpret this multi-trace map as using dualizable object and things like that? Or, cyclic monoidal category? Yeah, I mean, certainly people have thought about like, sort of the categorical versions of these kinds of traces. There's even been recent work on that by like Kerry Milkovich and Kate Ponto and that group of people. I'm not going to be able to make a concrete statement on top of my head. But yes, there are definitely people thinking about sort of categorical versions of this. That's going to be in my lectures tomorrow and then Thursday, so. Excellent, thank you. So, right, so let's unpack Goodwillie's theorem a little bit. So, negative cyclic homology. I'm not going to write out the full definition of what negative cyclic homology is for you. But the thing to know about negative cyclic homology is that it's another classical invariant from homological algebra. It's purely algebraically defined. You actually define it using the Hochschild complex, but you make it into a by complex and do some other things to it. But it's purely in algebra. Now, when I say rationally, it's a good approximation to algebraic K theory. What do we mean? Well, I could make it a little more precise. So in particular, he proves that if we have I and A, a nilpotent ideal, that when we look at the relative algebraic K theory of A rel I, once you tensor that with the rationals, then you get an isomorphism to this negative cyclic homology. So, in other words, when you tensor with the rationals, what happens? Well, you retain information about the rank of your algebraic K theory group, but you lose all information about the torsion. So what was happening in the 1980s is that various authors were developing tools to compute algebraic K theory by computing this negative cyclic homology. But what that's going to tell you is it's going to tell you about the rank of the K theory groups. So if you look at the 1980s, you'll see in the literature a lot of work on the rank of K groups using this kind of tool using Goodwill's theorem. So for example, if we look at our sort of sample calculation that we're taking with us throughout this historical story, there's a theorem of Soule's from 1981 where Soule computes the rank of the algebraic K theory of Z adjoin X mod X squared, again, the relative groups. And he proves that this is rank one if Q is odd and zero if Q is even. That was generalized in 1985 by Stasheff. And Stasheff looked at the K theory of Z adjoin X mod X to the M, so not just X squared, and proved that the rank of that is M minus one if Q is odd and zero if Q is even. So in the interest of transparency, Soule's proof actually doesn't use Goodwill's theorem, uses some other methods to get at this. Stasheff does use Goodwill's theorem in order to generalize this to Z adjoin X mod X to the M. So that's where we were in the 1980s, which was that there are a lot of calculations that were able to be done at that time of the ranks of K groups. But the big question then became how can we understand the torsion? So in 1990, Goodwill he gave an address at the ICM based on results of himself and Waldhausen and others. He conjectured that there should be what he called a brave new version of this story. So what do people mean when they talk about brave new algebra? Well, brave new algebra is the idea that we have all these classical constructions that we know and love in algebra. And perhaps we can define topological analogs of those constructions to translate those objects from algebra to topology. And those topological analogs may have some deep information in them that were lost in the algebraic context. So what Goodwill he conjectured is that there should be topological analogs of Hochschild homology and negative cyclic homology. And that those ideally would capture information about the torsion of algebraic K theory as well. So that vision became a reality not so long after. And the first one of these topological analogs is called topological Hochschild homology. So what's the idea behind topological Hochschild homology? Well, we wanna do this brave new algebra perspective. We wanna take something from algebra and ask for a topological analog of that. In order to do that, I need to think about what do they really need in algebra to define Hochschild homology? Well, Hochschild homology was an invariant of rings and to form that cyclic bar construction, I took tensor products of my rings. I didn't write it in the notation, but implicitly I was always tensoring over the integers. And once I had that, I was able to define the cyclic bar construction on a ring. Maybe I'll put a tensor in the notation, the note that I use tensor product. And from the cyclic bar construction, we were able to recover Hochschild homology. So that's what happened in algebra. Now I'd like to bring this into topology and how am I gonna do that? Well, I'd like to replace rings with a topological analog. And that topological analog is what we call ring spectra. So these are topological analogs that have sort of a multiplication in the sense of a ring. And tensor product then is gonna become smashed product in this topological setting. And so one way of thinking about what's happening is that I'm changing my ground ring instead of working over the integers, I'm changing my ground ring to work over the sphere spectrum. Once you've done that, if that all makes sense, then the cyclic bar construction, you can just define in the same way that we did before. So we can define the cyclic bar construction by now smashing together ring spectra, but it's the same definition of that type of simplicial thing. And from that cyclic bar construction, then I should get some topological version of Hochschild homology, which is called THH. Now the first person to sort of execute that idea was Bochsted. Bochsted did this quite a while ago. So he didn't have a lot of nice things that we have today, like nice categories of ring spectra with associated smashed product. So he had to do a bit more work, but he was certainly trying to execute this idea of translating this algebraic construction into topology. So topological Hochschild homology is supposed to be yet another approximation to algebraic K theory. So there is a map relating them, which is often referred to as the topological Dennis Trace. So that's a map from algebraic K theory to topological Hochschild homology. And if you're following really carefully, you might be a little bit confused about what I just wrote because I set up here that topological Hochschild homology is something that we do to ring spectra. And then right here, I wrote topological Hochschild homology of an actual ring. So what does that mean? Well, let me just note that I'm going to continue to write that. When we write topological Hochschild homology of an actual ring, what we mean, this is just notation for topological Hochschild homology of the Eilenberg-McLean spectrum of that ring. So to a ring, we can associate one of these rings spectra called the Eilenberg-McLean spectrum, but for, I don't know, ease of notation, we just write that as THH of A. Okay, so topological Hochschild homology, it is an approximation to algebraic K theory. It's better than Hochschild homology as an approximation. So we're moving in the right direction. But if you think about Goodwill's theorem, there was that negative cyclic homology that was quite close to K theory rationally. We'd like to have a topological analog of that as well. So in order to develop that, we need to notice a few things about THH. So topological Hochschild homology of A has an S1 action. For the same reason, Hochschild homology did, we're geometrically realizing something cyclic. And really, you can define this as what's called a genuine S1 spectrum. So we're gonna get more into that tomorrow of what that means and thinking about this as a spectrum with an S1 action versus a genuine S1 spectrum. But for now, let's just think it's a topological thing with an S1 action and we'll unpack what that really means tomorrow. So we're seeing again that equivalence. Now it turns out that topological Hochschild homology also has what's called a cyclotomic structure. So for today, I'm gonna black box this cyclotomic structure bit. Not because it's not important, but because I'm gonna really talk a lot about that tomorrow. We're gonna see today where it comes in and tomorrow I'll tell you really what it means. But it turns out that that's gonna be something that's really crucial to this story. So we'll talk about that in depth tomorrow. So how does that help us that topological Hochschild homology is this kind of equivalent object? Well, we wanna define a version of cyclic homology in this topological setting and that will be called topological cyclic homology. So I'm gonna tell you now the classical perspective on this which is due to Boxted, Shang and Madsen. But let me mention that there's some amazing recent work of Thomas Niklaus and Peter Schultzah that sort of reframes this definition of topological cyclic homology. And we're gonna talk about that tomorrow about the Niklaus Schultzah work and how it relates to this classical definition. But for now let me tell you how topological cyclic homology was defined classically. So what did Boxted, Shang, Madsen do? Well, they said, let's look at a cyclic subgroup of S1. So topological Hochschild homology has a group action and one thing you can ask about when you have a group action is you could ask about fixed points. So I could look at the CP fixed points of topological Hochschild homology. Now I might ask, well, how is that related to the thing that I had before I fixed anything? And one answer is, well, I could include the fixed points. So that gives me a map F, which is inclusion of fixed points. Okay, so I'm gonna draw a picture so we can keep track of what's happening here. So I had algebraic k theory and I had a map, the topological Dennis trace to THH. And now I'm saying, well, okay, let's look at the CP fixed points of THH. And I said, well, there's a map, this F map that relates the CP fixed points to the original spectrum. But actually I can keep going like that, right? I mean, I have a map, I could include the CP squared fixed points into the CP fixed points and so on. Now it turns out that there's also a second map, which I'll call R, which relates the CP to the N fixed points to the CP to the N minus one fixed points. This map, which we will discuss at length and tomorrow is called the restriction. And I'm not gonna define it for you today. There's not a nice one sentence thing way to say what it is, the same way there is for this F map. But the thing to know about the restriction for today is that it uses the cyclotomic structure. So in order to define this map, you need to know that you have that, thus far, mysterious cyclotomic structure that I referred to. So we have an R map like this. And Boxhacheng and Madsen proved that this topological Dennis trace, this map that I've got down here, actually lifts through all of these fixed points. And then the, so I give you this diagram on route to defining topological cyclic homology. So here's the definition of topological cyclic homology. So Boxhacheng and Madsen defined topological cyclic homology of A at a prime P is the following. I'm gonna look at all of these fixed points of THH. And then I'm gonna take a limit in some homotopy theoretic sense over all of these maps. So in other words, in my diagram, topological cyclic homology lives somewhere up here. And they proved that this topological Dennis trace lifts all the way to a trace to TC. So this is Boxhacheng and Madsen defined what we now call the cyclotomic trace from the algebraic K theory to the top, try that again, to the topological cyclic homology of A. Okay, and then the thing that makes this trace method so powerful is that in nice situations, TC is a good approximation to algebraic K theory. So there are lots of what we call comparison theorems that make that more precise. Like what do I mean when I say it's a good approximation? That's quite vague. So there are lots of different comparison theorems depending on the exact calculations you're interested in. But let me tell you one of them that's very powerful, which one way of thinking about this story is that we wanted an integral analog of that Goodwilli theorem to tell us about torsion and there is such an analog due to Dennis, Goodwilli and McCarthy. And what does their theorem say? It says, well, when you have a no potent ideal, then when you consider the relative algebraic K theory and compare it to, you can define relative topological cyclic homology, that that's actually in isomorphism. So that's the best kind of thing we could ask for that topological cyclic homology actually completely captures the algebraic K theory. So that may seem kind of abstract. You know, we went through equilibrium homotopy theory, we need to understand all these fixed points, but I claim that this can yield really concrete calculations in algebraic K theory. So let's first look at that computation we've been dragging through history and see what history tells us now. So where we left it in the 80s through Suley's theorem, we were able to understand the rank of those K groups, but not anything about the torsion. And so I, excuse me, I have a question. So the question is to go from TCAP to TC of A, did they use originally a fracture square definition? So I'm not sure. Yeah, right. Okay, so I've been careless maybe. So if you look at my diagram, I defined topological cyclic homology at a prime. And then in terms of the cyclic atomic trace, I gave you a statement that's sort of for all primes at one time. You can make a similar diagram where you do it for all primes at one time instead of just one prime, but then you have these restriction F maps anytime you have division and it's just messier. So I wrote it for one prime at a time, but you can reassemble the, like the TC of A at P essentially captures like what's happening at the prime P. And yes, you can reassemble the sort of integral information from that. So often we work one prime at a time in this area just because it makes the calculations simpler, but you can write the same kind of diagram and the same story to define it all at once. Okay, so what I was gonna do was I was gonna revisit that calculation that we sort of brought with us through the talk and through work of the Angle Bite, Myself and Lars Heselholz, we looked at this calculation again using sort of modern trace methods and equimering stable homotopy theory. And we proved that the odd algebraic A theory groups Z to join X, mod X to the M, rel the ideal generated by X. So Suley told us that those groups had rank M minus one or actually stash up told us those groups had rank M minus one, but they may or may not have torsion and we proved that indeed there's no torsion in those groups. And then the even groups for this same ring, Suley's theorem tells us that it's rank zero, but it may or may not have torsion and we proved that there is torsion and that the order of that torsion is MI factorial times I factorial to the M minus two. So this trace method approach can give you really concrete calculations in algebraic K theory. So that's the particular calculation that we sort of brought with us through the talk to see the development, but I wanna mention some very other very important calculations that were done via this method. So, shortly after the cyclotomic trace was defined, Boxsted and Madsen computed the algebraic K theory. Well, for primes greater than two, they computed the algebraic K theory of the Piatix. Let me just note that, I don't know how to note this, but in their original work, there was a conjecture for part of it and the conjecture was due to Saletus. So let me say that plus Saletus for a conjectural piece of it. Rognis revisited that for the prime two. So he did the analogous calculation at the prime two. A number of years ago now, Hesselhol and Madsen used this kind of trace method approach to study the K theory of local fields, some local fields. And they proved the Quillen-Lichtenbaum conjecture in those cases. So that was a while back. And many others. I mean, I could go on, there are lots and lots of important algebraic K theory calculations that have been done with this trace method approach. Let me just note that I've been focusing on rings, the K theory of actual rings, but these tools, this topological Hock-Schild homology, topological cyclic homology, are naturally defined for ring spectra. So you could also ask about more generally about ring spectra, not just rings. So for instance, there's a body of work by Osani and Rognis where they look at the algebraic K theory of topological K theory. So a little KU here is the connective topological K theory spectrum. Okay, so there have been lots and lots of algebraic K theory calculations using these trace methods and it's been very, very fruitful. So let me just close by mentioning what I wanna do tomorrow in my next lecture. So what I've said so far is that, well, the trace methods allows you to reduce your K theory calculation in good situations to computing topological cyclic homology. And that's good, but the claim is that topological cyclic homology is supposed to be more computable, right? That's only helpful if we can compute TC. And so tomorrow, we're gonna talk about, well, how do you actually compute TC? So how does one compute topological cyclic homology? What tools do you use? And we're gonna see that that leads us to really needing to understand this idea of what it means for a spectrum to be cyclotomic. So we're gonna talk about cyclotomic spectrum. And what we're gonna see when we do that is that these above calculations that I've mentioned, all these calculations that I've mentioned here, use some really serious equivalence stable homotopy theory to do those calculations. But there's beautiful new work of Thomas Niklaus and Peter Schultz, which gives us a new perspective in this area that allows us to actually move away from some of that equivalence. And so I'm also tomorrow, I'm gonna talk about the Niklaus-Schultz approach to cyclotomic spectrum and topological cyclic homology. And we'll see some specific calculations where that can be advantageous to think about it in that way. So that's a plan for tomorrow. But I think that was everything that I wanted to say for today. So I will stop there. Hey, thanks a lot, Tina. So now we have a question. So I'm sorry, I could not ask all the questions directly during the talk because there was too much. But if you have matured your question, you can now ask it to Tina and I will relay it. So there's one question about the comparison can you use about equivalence computation? So can you use top Tc, for example, to recover equivalence computation of a k-pherophonite field? Yeah, so when you, yes. So you can re-compute the algebraic. I mean, you can re-compute the, for instance, the k theory of FP completed at P, easily using, easily is the wrong word. It has been done using the trace method approach for sure. And that's written up nicely in some old notes of Eben Madsen. So he, what is it called? What are his notes called? Something like algebraic k theory and traces in that calculation is written explicitly. So yes, you can recover the k theory of FP completed at P using this kind of trace method approach. Okay, so another question linked. Do we have cases where Tc and k-pherophonite coincide? Yeah, so the Dundas-Goodwilli-McCarthy theorem is giving us cases like that. So for instance, one of the calculations that I, that I, right. So I mean, so that the situation it's capturing is sort of things like you see in this theorem here where you have relative algebraic k theory, relative sum nil potent ideal. And in those cases, the Tc and the k theory do coincide exactly. There are other theorems. I'm telling you one Dundas-Goodwilli-McCarthy theorem and there are other theorems due to people like Hesseholtz and Thomas Geisser. And I'm gonna forget if I try to list everybody, but there are lots of comparison theorems in specific cases. And what usually happens is that depending on what kind of calculation you're doing, you may, you look for maybe a slightly different comparison theorem. But the hope is that you have a comparison theorem that either tells you that they coincide directly or is gonna give you some information about the relationship between the two. But you probably have to be complete, right? K theory, because if you look at the different prime it's a different story. Yeah, I mean, so Dundas-Goodwilli-McCarthy, so the original Dundas-McCarthy theorem, involved a peak completion and then there's, they've sort of their most recent version of this theorem that I mentioned up here is an integral theorem. But maybe to get to your point of what you were mentioning about the peak completion is that, lest I've left you believing that trace methods is gonna solve all of our problems. Let me make a comment, which is that, you look at this, I claim that many years ago now, Baxian and Madsen compute the algebraic K theory of the Piatics. So you might look at that and feel very optimistic then about the algebraic K theory of the integers, right? Like surely that should be closely related to the algebraic K theory of the Piatics. But unfortunately, algebraic K theory doesn't behave nicely with respect to peak completion the way topological cyclic homology actually does. So when they compute this, they look at the cyclotomic trace to topological cyclic homology and that cyclotomic trace is an isomorphism. Now it turns out that topological cyclic homology that for TC, this kind of like peak completion inside and outside behaves nicely. And this is the same as the topological cyclic homology of the integers P completed. So one might hope, well, I have a map from the K theory of the integers to TC. So I have a map like that, which is the cyclotomic trace. And so you might look at that and think, well, then wouldn't it be nice if I could say something nice about how these two are related to each other? But K theory doesn't share this property that TC does that this peak completion inside and out gives you this isomorphism. So you have to be a bit careful. So for instance, Bach said, Chang's Madsen's result is not gonna tell us about the K theory of the integers. And the K theory of the integers is in part, I'm not gonna say intractable, but part of why it's difficult is it's not very approachable via this method because in that case, I've highlighted too many things. In that case, this cyclotomic trace map over here is not an equivalence or something that we can really understand very well. Okay, so next question is, is there an extension of TC, THH two schemes? Or nice? Yeah, yes. So you can study topological cyclic homology and THH for schemes. I don't know that I have a lot of good things to say, tell you about that, but I will give you reference, which is that work of Lars Hessel and Thomas Geisser, for instance, looks at how to study these things in the algebraic geometry setting as well. So next question is in the theorem on galvite and elselolt. And it is that Geisser or, no, on Godvili. Oh, it's me. It's a big, sorry. Is there, do you know the group structure on the torsion rather than the order? Right, okay. So when we look at this theorem that I mentioned of galvite and elselolt, we compute the order of these groups and even dimensions, but I haven't told you what the group structure is. No, we don't know the group structure and using the sort of methods that we use, that question is very intractable. I'm trying to see if I can explain why using what I've said so far. So it turns out that these calculations that I've done here, maybe it'll be more clear tomorrow why this is the case, but these calculations are gonna reduce to some equivariant homotopy calculations in ROS1 graded equivariant homotopy theory. And those calculations that go into this K theory calculation with current technology, there's no expectation that you'd be able to recover the group structures. And so we're able to like do some tricks to get the order of the groups, but we don't know the group structures except in low dimensions, like I couldn't do it off the top of my head, but for some small values of I here, you can explicitly say what these groups are, but in some general sense, no, we don't know and it's not so tractable. Okay, next question. So we still have four questions. So is it, do we know something more about the structure of KQZX mod X X2VM comma X for Q even? Yeah, I think that's the same question. I mean, I think that what that question is getting at is about whether we know the group structure here and I know, except in low degrees. Okay, so next question is about the action of S1 on BCCI A, can you define this by using the trace map? So the way to see the action on the cyclic bar construction, let me back up a little bit and see if we have a picture of the cyclic bar construction. Okay, here it is. We have this picture of the cyclic bar construction. I said sort of mysteriously, well, it's a cyclic object and therefore it has an S1 action like magic. So the way to think about that is what a cyclic object gives you essentially is that like in simplicial degree, this is, there's like an off by one thing that's a little weird. This is simplicial degree zero and one and two. In simplicial degree one, you have a ZMA2 action, which is like permutation of those, these tensors. In simplicial degree two, you have a ZMA3 action. In simplicial degree three, you have a ZMA4 action, et cetera. And what's happening with this S1 action on the geometric realization is essentially saying, well, all of those group actions assemble in some nice way into an S1 action on the geometric realization. So that's due to Alan Kahn and sort of the theory of cyclic sets. So that's how to think about what that S1 action is coming from in the cyclic bar construction. Okay, so that question, but I still ask it, do we have an over-interesting application of TC, THH, other than two algebraic K theory? Yeah, that's a great question. So I would say, up until recently, if you talked about TC and THH, people would usually say, well, it's mostly as a means to compute algebraic K theory. That's like one way of thinking about what those are. And that's certainly still true. But arguably right now, people are starting to see a lot of interesting applications, particularly of THH on its own. So there's this recent work of Bhargav Bhatt, Matthew Morrow and Peter Schultzah, where they look at connections between topological Hock-Schultz homology and piatic Hodge theory. And I maybe won't say so much about that, but maybe to say one thing about it is that I mentioned briefly that there's a way to put a filtration on algebraic K theory so that you get a spectral sequence relating motivic homology and algebraic K theory. And Bob Morrow and Schultzah look at, well, what happens if you try to put such a filtration on topological cyclic homology and topological Hock-Schultz homology? And they're able to get relationships between those objects with other theories such as crystal and homology. So yes, in recent years, there's a lot of interest in particularly THH as it's sort of own invariant to study. But historically, they were certainly like means to understand algebraic K theory, but that we've seen a lot of new developments in the last few years about that. Okay, so still three questions because new questions arrive. So about computation. So I have one, is it easier to compute Tc? For example, Tc of a ring, for example, A and Z mod P and things like that. Is there one K theory for example? I may interpret the question, but yeah. It just depends. Sometimes yes and sometimes no. I mean, like so for the integers, yes, like topological cyclic homology of the integers is definitely more accessible than the K theory of the integers because as we can see, sort of, oh, I don't know where I was saying it. Maybe it's, oh, here we go. Topological cyclic homology of the integers was actually computed by Bakshad and Madsen many years ago, but the K theory of the integers is still unknown. So yes, in some cases, Tc is definitely more accessible than algebraic K theory. And so there are certain calculations where the issue is not having a good comparison or not understanding the comparison between K theory and Tc, that's definitely something that happens. But another thing that happens is that you just can't compute Tc in the first place. So depending on a specific calculation, there are sort of different roadblocks that come into play. Okay, so some kind of questions. So is it still hard to compute K theory of Z mod P to the N? So even using the DGM theorem, so I guess it's Dundas, Goodvely, and I don't want to get the last name wrong, so. McCarthy. Yeah, only present ideal, techniques like that. Yeah, so the K theory of Z mod P to the N is still open. Certainly people have tried it. There's work of Morton Brun, and well, he is more working on Tch of Z mod P to the N. And be like Engeltweid has thought about this and other people I think are thinking about it now using sort of new technology. So people have tried it with trace methods and ideas have included like trying to filter it somehow and access it that way, but it's not, it's certainly, it's open, it's not easy using this approach. And I think a lot of people have given thought to whether it's possible, how to make it possible. So there is a question about computing K theory of Z X Y, Z bracket X Y mod X comma Y. Yeah, so the question's about the K theory of the coordinate axes. So Z join X and Y mod X Y. And the answer to that of whether this is computed is yes, the K theory of this was computed by V Black Engeltweid and myself. And it is similar to this calculation up here in the sense that, let me get this right. In the odd degrees, we compute the groups exactly. And in the even degrees, we compute the orders of the torsion groups for sort of similar reasons that I mentioned in that other calculation. But yes, these are understood. And also, if you work over like a field, perfect field of characteristic P that was understood prior by work of pestle holds. And tomorrow we'll touch more on like, why a little bit on why we consider these kinds of rings and how that would be approachable at all using this method. So last question this time, but I'm very happy that we have so many questions. So first of all is fair. So can we understand TCA, P in relation with THH as K theory with P completed coefficients relates to K theory with integral coefficient of ZP, let's say. Is it the analogy, but? I'm sorry, I'm reading the question. I'm trying to understand what's asking. So maybe we can answer the second question. It should be easier. So there is a construction of TC of A as THH of the island of McLean spectrum, HA. Can we apply the same construction to KA? No, so the construction where you take topological hoc shield homology of A and build from it topological cyclic homology of A relied on the fact that on topological hoc shield homology of A, you have an S1 action. We use that in a very important way further that it was a cyclotomic spectrum. K theory doesn't have an S1 action. So it doesn't make sense to talk about like taking those fixed points and building that tower in the same way that you do for THH. So it's just the, right? You just can't do that construction at the level of K theory. So maybe if you have a question, we will continue in the comments after another question I did. So let me, so we talked about negative cyclical homology. Is there notion of non-negative cyclical homology? Yeah. Negative topological cyclical homology. Right, okay. So yes, yes, I'm all friends. Okay, there's hoc shield homology, which I defined for you. There's negative cyclical homology, which I didn't define but mentioned. There is cyclical homology, and then there's another one called periodic homology. So the way to think about this is that hoc shield homology, I defined for you the chain complex that you use. You can extend that to a by complex. And cyclical homology is what you get when you only consider the positive part. Negative cyclical homology is what you get when you only consider the negative part. And periodic homology is what you get when you consider the whole thing and take the sort of total complex of these things and take its homology. So we have algebraic theories of all of those. And then your question about topological versions of that is an excellent one. So for hoc shield homology, we have topological hoc shield homology. For periodic homology, we do now just recently people have started to consider something called topological periodic homology, which I will define to borrow. And then for negative cyclical homology, it gets a little confusing because I said, well, topological cyclical homology is like our analog of negative cyclical homology. And that's true. But if you look at the actual definition of negative cyclical homology, there's a more direct topological analog, which until recently people didn't really consider because it doesn't behave well. It doesn't do the thing that you want it to do. Turns out that it plays an important role in this work of Nicolaus Schultz. So that's what we call topological negative cyclical homology or some people negative topological cyclical homology. I don't know, there's no agreement there yet. And again, that I will define for you tomorrow. So yes, there are more algebraic theories here and they have topological analogs which are interesting and will come up when we talk about how to really get our hands on TC. Okay, so I think we have finished the question now. So thanks a lot, Tina. Again, we will hear you the next talk in the afterwards. So let me look, tomorrow you'll have your second talk and then on Thursday.