 Okay, there's this traditional moment where the grateful audience thanks the organizing committee. But as a member of the organizing committee, I wanted to short circuit that moment by saying just a word about what it was like to organize this conference. Organizing this conference meant, you know, we said to Paul, you know, something, and then we would get a detailed three-page, Paul Zidle here. We would get a detailed kind of three-page email thinking through all of the tiny details and explaining to us how he met with the Italians and discussed the details and this and that. And we would send emails that said, uh-huh. So anyways, so as the organizing committee would like to take this opportunity to very gratefully thank the spectacular organizer, Paul Zidle, before he skips town, think about also summer school participants. He also organized the summer school, so it's all expressed our thanks to Paul. And I will now turn this over to a different Powell. Thank you. As you can see, I will be chair this morning, but please do not be alarmed because I got specific instructions about the duration of the talks. So I would like to introduce our first speaker of the morning, Dave Gabbai, who will talk about knotted three bolts in S4 and knotted three spheres in S1 crosses three. Thank you, Powell. Well, it's really great to be here at this meeting in person in honor of Tom. And I should say that I had had the pleasure of knowing Tom during a period where before he was famous, by which I mean, I mean, I knew Tom at a time when amongst experts, I mean, experts knew that he was already doing great work, but his fame hadn't sort of spread to the sort of broader mathematical community. So I'd like to just tell just briefly just two sort of anecdotes from that period. It's been a long time ago. So if the events didn't transpire exactly as I say, well, they should have. And so years ago, I was at Caltech on the faculty, and Tom was there in some type of postdoctoral position. And that full semester, I went to a couple of different meetings, and someone would come up to me and say, Tom Rovka is in your department. He's doing just amazing work. You should hire him as a senior faculty member. And he said, like, Tom who? But then another person would say the exact same thing. And this happened like three or four different times. So at the time at Caltech, there was basically two people that were like the driving force setting the direction. So I went over to one of them with this wad of paper with Tom's papers, and I said, well, we should seriously consider just making Tom an offer. And so I handed these papers over, and he sort of flips through and says, well, these are just his preprints. Where is his published papers? And I said, well, there really aren't any. And then he sort of flips through again, and he says, there's only two preprints here. And I said, yes, but they're good preprints. And anyway, this guy was, he's very well connected. I think he was already sort of clued into Tom, actually before I was. And it's very supportive through the whole thing. And the fact is that the case just sailed through the department and the administration. And okay, so let me tell you another little story, which is, I guess around that time, there was some special lecture at Caltech. And well, by Caltech standards, there's this huge audience packed into this lecture hall. And sometime in the beginning, towards the beginning of the lecture, the speaker just sort of throws out a problem to the question to the audience. And Tom sort of immediately, so just casually just gives the correct answer. And the speaker seemed a little bit surprised, but okay, and then a little time later, the speaker asks another question sort of with a tone of, well, I don't expect anyone to answer to this, but if they try, they'll probably miss the subtle point. Which point, Tom just casually just nails the question. And this guy is just totally taking it back and after a moment says, like, who is this guy? And by that point, Tom already was famous at Caltech, so this cry went up from the audience and said, well, that's Tom Rovka. And there's sort of like this no sort of recognition from this person, but on the other hand, there are no more questions. So, right, so, okay, so I want to tell you about nodded three bulls in S4 and nodded three spheres in S1 cross S3. So this lecture is about co-dimension one nodding in four manifolds. So the big question is in the four sphere, is there like a smoothly embedded co-dimension one nod? Is there a smoothly embedded three sphere in the four sphere which is not topical, not smoothly standard? It's every smooth three sphere in the four sphere smoothly standard. Of course, that's sort of probably everyone knows and well, I'm sorry, I can't say anything about that, but closely related problems are the problem of nodded spheres in S1 cross S3 and nodded bulls. So to see this, if you have a smooth sphere in the four sphere and you just take a point in each side, then the compliment is just S3 cross R. So you could just view the spheres living in S3 cross R or even S1 cross S3 and conversely, if you have an interesting sphere in S1 cross S3, then you could sort of lift to the infinite circle cover and view that sphere sitting in S1 cross R, which is sitting inside of S4. So the question of nodded three spheres, nodded separating three spheres in S1 cross S3 is very closely related to the studying nodded spheres in S4. And so the difference is that you might have a nodded sphere that becomes unnodded when you lift some finite sheeted cover. But nevertheless, this thinking about it in this way just sort of gives you sort of an approach to either constructing interesting examples or somehow coming to grips of what are sort of phenomena that have to be dealt with in addressing this question. And another closely related thing, which I'll sort of give a definition in a moment is this question of nodded three balls in the force sphere. So, right, so what do I mean by a nodded three ball in the force sphere? So what I mean is that here we are in the force sphere and here's say the standard two sphere and the standard two sphere bounds the standard three ball. So we just decide on what those things are. Then we say a ball is nodded if its boundary coincides with is the standard two sphere, but this ball can't be isotopped to the standard sphere by an isotopey fixing the boundary point-wise. So here's, well, I mean you could make this definition in all dimensions and so here's, well, I mean the classic dimensions of two and three, it's well known that there's no nodded balls in one balls in two space or two balls in three space. And so what's the question, what's the relation between this question of nodded three balls in force sphere versus nodded three spheres in the force sphere? And the fact is if you have a nodded three sphere in the force sphere, then you can make that nodded sphere look standard at a neighborhood of some ball. And so the complementary ball would be then a nodded three ball in the force sphere and conversely, if you have a nodded three ball in the force sphere and you happen to find another ball which shared the same boundary and its interior is disjoint from the first one and the second one was a standard ball, then you could put the two things together and you would construct a nodded three sphere in the force sphere. So that's the relation between nodded balls and nodded spheres. And I should say that the way I've stated this definition of nodded ball of K ball and K plus one space, you could just as well sort of ask this question about balls of bigger co-dimension. So you could ask, well, if you have a standard circle in force space and the standard two disc and now if you have another two disc with the same boundary are those two discs isotopic from one to the other fixing the boundary point wise? And, well, the answer is yes, that's what, I proved that some time ago, that's actually that's a form of the four dimensional light bulb theorem. So the light bulb theorem is you could take that as sort of a warm up to this question of nodded balls. It's just the same question, one higher co-dimension two instead of co-dimension one. Now, I should say that there was sort of, it was known, well, 50 years ago that there's in high dimensions, there's nodded, well, nodded five balls in six space and nodded six balls in seven space and so on and so forth. Actually, that's due to Hatcher and Hatcher-Wagoner and actually I don't think that they had this concept of nodded balls but they proved something amazing result which as you'll see in a little bit is equivalent to this question. So it's fair enough to attribute this theorem to them. So, right. So I've been thinking about this for a while and I first thought that well, that there weren't, every three ball was standard but around 2017 I sort of convinced myself that there were sort of nodded examples and that's sort of the main content of today's lecture which is that there really are nodded three balls in four space and that's due to myself and Ryan Budney and independently Tariuki Watanabe. So the classic dimensions, arcs in two space, two discs in three space. Okay, so this lecture is about three balls in four space. This question, are there nodded four balls in five space? Well, Ryan and I think that there aren't any and sorry, that there are nodded four balls in five space and we have sort of explicit example which I'll show you at the end. There's time and then the higher dimensions, that's due to Hatcher and Wagoner. So, right. So, okay, so this lecture is about nodded balls in three balls in four space. So, but sort of a counterpoint is this basic fundamental theorem of Serf and Palais from over 60 years ago and this is a theorem that Bob Edwards calls differential topology 101 and their theorem is that, well, modulo the obvious necessary conditions, two k-dimensional balls in some four manifold, any two k-dimensional balls in a four manifold are ambiently isotopic. So, I mean the obvious condition is they have to be in the same sort of component and if they're co-dimension, if they're balls of co-dimension one, well, if they're co-dimension zero, then they certainly have to have the right induced orientation. So, and this is sort of a nice exercise in differential topology and you could see that here's sort of the key idea. I mean, you could think of this as sort of telling you how to do an ambient isotope of a germ of a ball in n-space. I mean, well, here's the formula and you could see that just there's a smooth isotope of taking this embedding to one which is linear and then you can sort of straighten it up. So, that's a funny thing is that just the standard theorem is that any two balls are smoothly isotopic if you're allowed to sort of move the boundaries around but once you fix the boundary, then that's sort of a different question. So, here's three equivalent formulations of the same thing. First glance, these ideas seem very different but just basically relatively elementary arguments show they're the same. So, you can see like condition one, knotted ball versus diff of something. How do you relate sort of embeddings versus difthomorphisms? Well, well, the fact is suppose you have some ball in the three-sphere with the standard two-sphere boundary and so surf and palais tell us that there's sort of a, an ambient isotope of three-space taking one to the other. So, the time one map of this isotope is a difthomorphism of the four-sphere which would take this knotted ball to the standard one. So, up to difthomorphism, thanks to surf and palais, there's just really only one knotted ball. This question could also be viewed in S1 cross three ball. So, if you have three ball cross S1, then we have the idea of the standard three ball, point cross three ball. So, we'll say a three ball is knotted if it say it has the same boundary but it's different. So, and well as we all know that if you have a standard two-sphere and four-space, then if you remove a neighborhood of that sphere, then you're left with S1 cross the three ball. So, in this way you could translate questions about S1 cross the three ball into questions about knotted three balls and four-space. And, as I explained, the surf palais tells you how to relate balls versus difthomorphism. So, if you have a difthomorphism of the four-sphere fixing the two-sphere that takes one ball to the other, that induces a difthomorphism of S1 cross B three, which fixes the boundary point-wise and takes sort of one ball to the other. And actually these things are groups. I mean, you could actually compose them. You could certainly see that at the level of difthomorphism. So anyway, these are three useful ways of thinking about the same issue. And, in this great work of Hatcher and Hatcher Wagon or from 50 years ago, what they did, they computed pi zero of difth of S1 cross the n ball where n was, say, bigger than equal to six, difthomorphism's fixing the boundary. Okay, so, right. So, I wanna tell you a few basic facts before we maybe dive into some, yeah, Danny. Oh, so the point is, this theorem here is three is a variable. Three is an integer. And four, yeah, yeah, yeah, anyway, thank you for saying that. You know, this is a theorem that's true in all dimensions. So, three could be, three is any positive integer and four is the next integer after three. Just the way I'm wired, I just like to think of things sort of concretely, but anyway, thanks for bringing that up. And so, here's actually a very useful vibration. I mean, this idea of vibrations among, you know, relating embeddings and difthomorphism's various things goes back to surf around 1960. And a very useful fiber bundle is this one that I have up there. And the idea is that if you have a diff, so the little zero, diff zero, means difthomorphism's homotopic to identity. So, if you have a difthomorphism of s one cross s three, you could ask, well, where does the s one go to? So, that's an embedding of s one and s one cross s three. And the little zero means the embedding is in the right homotopy class and the orientation's sort of pointing up. And the fiber of that is difthomorphisms of s one cross s three, which fix, fix this standard vertical circle. And if you apply the homotopy exact sequence corresponding to this bundle, I mean, the last terms are, you know, as shown here. So, you could see pi one of the embeddings to the circle go into pi zero of diff of s one cross b three, which goes into pi zero of diff of s one cross s three. I mean, just actually a little bit of translation going from one to the other using uniqueness of regular neighborhoods, whatever. But, right. So, here's the theorem which we prove is that if you look at pi zero of diff of s one cross s three, well, modulo difthomorphisms that are sort of supported on the four, on some little four ball, that's an infinitely generated group. And similarly, the difthomorphisms of s one cross b three which fix the boundary, modulo difth of four balls, that's also infinitely generated. So, by this correspondence, I just state, what, diff zero, isn't diff zero connected? That's a really great question. And that's what, so diff, so surfs great, well, one of surfs great theorems is diff zero of the three spheres connected. Well, what about s one cross s three? That's, that was, that's what, oh. So, okay, so, so diff of s one cross b three fix the boundary, left. Pi one of embedding zero, oh. So, so diff zero, actually, you also bring up a good question because I use this expression diff zero actually in papers in two different ways. One could be, you know, diff zero can be viewed as diff, the identity component of, of, of, of diff, or it could be difthomorphisms that are homotopic to identity. So, so thank you for pointing this out. So, so here diff zero means difthomorphisms homotopic to identity as opposed to identity component. So, right, so you almost killed the lecture. Thank you. So, right, are any other questions? So, okay, so here's, so you might, so, so, so our stuff is extremely explicit. And, and here are, are pictures of, of knotted balls. So here we're, it's, here we're, we're, we're, we're viewing this pic, viewing these balls in S one cross to three ball. And so, so look at the picture on the left. Actually, the conjecture is this, this ball is, is knotted. If you look at an early version of our paper, maybe early two, first two versions, we, we claimed that it was, that it was knotted, but it's, it's, it's, there's a mistake in that. And so this, it's, it's conjecture. And as far as I know, Tadiyuki does, you know, it's, it's, his, his technology doesn't, doesn't address this. But, so, so how do you sort of parse this picture? And so, so in dimension three, we know this idea of surgery on, on a three manifold, right? You drill out a, a, a, a solid torus and re-glue. But as four manifold people, we know that this could be done sort of ambiently, that, that attaching, the effect of attaching like a two handle onto a four manifold with boundary, it changes the boundary by, by, by, by surgery. So, so we can think of surgery, sometimes we can think of surgery sort of ambiently. And that's, that's what we have here. So, so we have like, like delta zero, that's, that's the standard three ball in S one cross B three. And sigma is a two handle. So we could modify this, this ball by, by attaching a two handle, sigma to, on the, on the, on one side, and then we could attach a two handle towel on the other side. And, and, and, and, and if these two handles are zero framed and the attaching circle is the hop flink, then at the three dimensional level, we're just doing zero surgery on the hop flink, which just takes the three ball to itself. But, but at the, at the four dimensional level, we could, we could see that this, this changes the embedding. And, well, well it's a question of the, your perspective here, but one of the two handles looks standard, this guy's sigma, and the other two handle sort of is linking through sigma and, and, and links around itself. And that's tail. So these, so this, this is description of, of this conjectural knotted three ball. And, and the one on the right, well it's, it consists of three sets of embedded surgery. So you, so at the three dimensional level, you're doing, you're doing surgery, zero surgery on, on three hop flinks that are just totally, totally separated from each other. So, so at the three dimensional level, you're not changing the manifold, four dimensionally getting this complicated embedding. Yeah, so, so, yeah, so, so if you have a four manifold with boundary, attach a two handle, it changes the boundary. So, so strictly speaking, you know, this delta zero is just a three ball by itself. But think of it locally as like a three ball across the interval. And when you attach sigma on one side, you're, you're changing this three ball by, you know, as you say, adding s two cross s two factor. But then once you have that, then look at that three manifold, look at that the resulting three manifold, thicken that up across the interval and attach two handle now to the other side. So, so, so that sort of at the three dimensional level sort of kills the, oh, you know, exactly. We're constructing a co-bordism between, you know, the ball connected some s, well, s two cross s one with, with itself. Yeah, thank, thank you. Okay, but one thing I'd like to mention is this picture really shows why I think four dimensional topology is smooth four dimensional topology is really challenging. I mean, you didn't have to come to Italy to hear from Dave goodbye that two plus two is equal to four. However, that's, but if you ask yourself, well, let's try to understand sort of how to, you know, work with embeddings of three dimensional things sitting in four space. The point of this picture, well, one point of this, you should get out of this is that even though this is a three dimensional thing, there's parts of it that really look two dimensional. And I, you know, I've, so my word for this, these are what I, this is what, this sign is what I would call an anilini disk. So, so anilini, that's, it's, it's, these are like, I might be saying the word wrong, sorry if you're Italian and I'm not Italian. But anilini is a, is a, is a pasta, which is a tiny round circle. So, so now we have like Sigma, that's a, a two disk. But, but at the, what's really looking, but when you, at the three dimensional level, what you see is really like the unit normal bundle of, of, of Sigma, that's really, you know, D two cross S one. So when you're attaching this handle, you're, you're replacing the solid torus with this D two cross S one. But, but it, but it looks, you know, visually, just, just, just two dimensions. And, and, and the same thing this, this towel looks, looks two dimensional. So, so, so the point is this, this, this three dimensional embedding to our eyes looks, you know, parts of it looks very two dimensional. It's, it's like the same, you know, one dimension lower if you have, you know, a surface. And you can think of, you know, this nice surface or you can think of like two pieces of surface, you know, connected by an extremely thin tube. And so, so, so the surface really is two dimensional, but there's one spot where it looks, looks one dimensional. So, so the reason why this, this interesting is, you know, in co-dimension one in three manifolds, you have surface in a three manifold, something like mean curvature flow or some two dimensional, you know, some, you know, topological analog of that, extremely powerful for, you know, straightening out surfaces. But if you, if you did like mean curvature flow to, to this, this thing, well, these tiny little circles immediately would sort of, you know, become singularities. And so this, these circles are just, just, just contract out. And, and okay, but if now you did some type of, you know, mean curvature flow, but somehow you, you, you just constrain it by putting in some filling. And, but, but still these are, you know, at the level of the surface, this tail just wants to rip through the sigma. So, so this was to sort of, you know, illustrate this idea that, that if you, you know, can, can, you know, control co-dimension one embeddings, then, then there has to be some type of global understanding. This, you know, local, local moves are going to be problematic. So, right. So, okay, so I've got 10 minutes, 11 minutes to, well, to just explain, okay. So this, that was the introduction to this lecture. No, no, that's, that's right. 45 minutes, 45 minutes. And, you know, pool, it's, I'm very happy to sacrifice these few minutes to pool. But, but the question is, how do you, okay, so, but, but the idea is, you know, just a huge number of detail, but the idea is sort of like really, really simple, which is, you know, how do you construct sort of candidates for interesting balls? And it's, it's really just a matter of like pie, by which I mean, if you, you know, making a pie and you, you have this, you know, the, the vanilla and the chocolate, you put it on top, and now you put the, put the, put the spatula in. So that's the starting point. And now you just like move the spatula around and then bring the spatula back to where it started. So now you've done basically a difumorphism of the complement of the spatula. And that's, that sort of gives sort of a candidate for, for an interesting difumorphism of the complement. So what does this have to do with knotted balls? Well, you saw there was this exact sequence where one of the terms of pie one of embeddings of the circle in S one cross S three, somehow induce difumorphisms of S one cross B three. So the spatula is the circle. And if you have some like, you know, you know, loop in the embeddings and you bring the, the spatula back, well that, that induces sort of difumorphism of, of S one cross B three. I mean, something we're all sort of familiar with, which is like in one dimension lower point pushing, which, which Joan Berman for me was, which is actually, that was her PhD thesis. So, so right. So there's a map from, from pie one of the embeddings to pie zero of, of diff. And right. So, so anyway, you know, the embeddings, pie one of the embeddings of the circle in, in S one cross S three, backs 50 years ago showed that was infinitely generated and, and a row in Zimek gave a sort of more modern proof. And here, Brian and I found sort of explicit set of generators, I mean, whatever, whatever that means. I mean, somehow you're, you're, you know, you have like an arc here, this, this, this, this pink thing. Here you have a picture in three space and this, this arc it's going in the past and the future. And, and on the other hand, here's an arc in, in three space. And now you can just spin this arc in three space around this, this point. And so that's, that's a, that's a loop of, of an arc, but, but the red thing is, is going in the past and the future. So anyway, you know, it's, I don't have really time to explain any of this, but, but that's either the key thing. So these, this, this thing here, these are sort of, this data two, data three, data four, these are, so generators for, for embeddings of the circle and, and, and S one cross S three. And, and now there's, well, this whole story about how, yes? No, no, they're, they're, no, no, these, these actually generate. They're, they're actually embeddings of pi one, up to sort of like a trivial, well, just a translation. All the other embeddings, it's, it's, it's, this group is infinitely generated and these are the free generators. And so, so now there you have to get to this whole process of, of translating a, you know, this, this, this, going from a loop to a diffeomorphism and that's what we call barbell implantation. And so, so this picture on the left shows an embedding of, of, of two, two spheres connected by an arc. And, and so, so it turns out there's this, this really interesting, extremely elementary diffeomorphism of, of, of a neighborhood of, of this barbell. I mean, neighborhood barbell is simply boundary connected sum of S two cross D two with itself. And, and so there's a very simple diffeomorphism of S two cross D two connect, boundary connected sum with itself which fixes the boundary point wise. And if you have a diffeomorphism of that and you have an embedding of that in some other form manifold, then you could push forward this barbell diffeomorphism and, and, and anyway that, that's, that's what we call barbell implantation. And so, so we explicitly, we can translate sort of this, this loop of, of, on, on the into diffeomorphisms. So, oh, anyway, this is just a question which is, you know, here's a, here's a, that's called a conjecture that, that diff of S one cross S three are generated by barbell implantations. And, and actually all the diffeomorphisms that Tadiyuki constructs, those are, those come from barbell implantations, composition of, of such implantations. Okay, so, well, I don't really have time to explain this, but the question is, suppose by some miracle you have, you know, an explicit thing and the question is, how would you actually prove that it's, it's actually non-trivial. And, well, here's sort of like a warm up which is, here's this three manifold which is, which is just the, the solid torus minus an open ball. Now you could cross that with the interval. And, and here's the two disc, this thing D zero which is the standard vertical two disc. And, and here's another two disc D one which is gone by just tubing. Well, this, this thing P is a two sphere and I get attached a two disc, a tube from D zero to P by, it's sort of, so it just goes through P, goes around this element, links through P, goes around the element of fundamental group and connects to P. And, so, so it turns out there's a diffeomorphism of this four manifold, homotopic identity which takes one, one two disc to the other two disc. But, but in fact, these two discs are not isotopic. I mean, they're not isotopic, you know, fixing the boundary point-wise. And anyway, the point is that, that a two disc is a two disc, but a two disc is also a one parameter family of one discs. And, and so, so if you have a, you know, so embedding of the two disc, that gives you an element of pi one of the embeddings of the one disc in this, in this four manifold. And, and again, this goes back to DAX, same paper that I mentioned before, is that using his sort of methods, you can actually show that these things, these correspond to two different elements of pi one of the embeddings. And, well, I don't have time to explain this picture, but this, this actually also gives construction of a knotted three ball in, in this four manifold. So, right, so anyway, I'll just flip through some pretty pictures, but, but what, what, but what we do is, is now, now if you have a three ball, and say S one cross B three, a three ball is a two, you could think of that as a two parameter family of, of intervals. And, and so, so an embedding of the three ball gives an element of pi two of this space of embeddings of the interval into, into S one cross B three. And, so, so, so anyway, to the, the point is we have to develop this all, this geometric theory that we can actually, to actually like work with these things. And, and we found these generators called GPQ. And, and it was, it was just a tremendously long, well, to me, to us, really long calculation. And the end we showed that, well, somehow that, well, we just got all these, all these elements of pi two are trivial. So, right, so that was the bummer. And, okay, so, so, so like, well, what do we do then? So, but, but then we realized that, that the picture on the left corresponds to a barbell implantation coming from, from one of these elements of pi one. But we can somehow like not the, not the, the implantation and that changes to the calculation by such and such. And we, well, we, we, we actually proved that, that, that these, these implantations correspond to, to sort of independent elements of, of pi zero. So, well, diff of S one cross B three. And, well, okay, then that's a whole, starts a whole nother story, which is, well, okay, we get these things, they seem to be independent in terms of these GPQs, but, but then we have to actually show this and how these, and anyway, we know GPQ as some relation. And they seem to be independent module, this relation, you know, you know, after including this relation. And anyway, then we have to, anyway, there's this whole thing, you know, DAX sort of proved this, this great theorem in this day. I mean, on understanding embedding, actually he was about, you know, if you have a one manifold mapped into another manifold space of embeddings and provided there's, you know, enough co-dimension, you can say some things. And, and it was relevant in the case of maps of the interval into four manifolds. But, you know, there's this whole fantastic branch of mathematics, you know, developing theory of embedding spaces and, you know, Tom Goodwilly and John Klein and Michael Weiss. I mean, developed this fantastic theory and then, you know, Dev Sencha sort of progressed in a certain way. Anyway, using, using this whole stuff, we could show these things are non-trivial. And so, so anyway, so that, so, so, so these things are these, so, so, so anyway, you know, using their theory, we could show these, these represent non-trivial elements of pi two embeddings of the interval and S one cross B three. So, so here this picture is, is supposed to explain why the bad thing, which is all these things are trivial, is now the good thing. And the bad thing, and the good thing is, is that, well, you know, just the stuff I just muttered about just moments ago was that we had this map from pi zero of diff of S one cross B three fixing the boundary into pi two embeddings of the interval. And, and we showed that there's some elements that were, you know, that, that not trivially. On the other hand, well, the bad thing was that, that when we, when we looked at this, this map five prime on different morphisms coming from loops of embeddings, they all map to zero. So, so we know from, you know, basic algebra, that means that, well, I mean, you know, from the exact sequence, you know, that means that we get an induced map from pi zero of diff zero S one cross S three into pi two of the embeddings. And so, so our, our elements that we construct are, you know, we actually can view them as non-trivial elements of pi zero diff S one cross S three. Whoops, two minutes over time. So, right, so that's, that's that. And so anyway, here's, here's a conjecture for a knotted four ball in, in a five space or viewed in S one cross B four. Two spheres link in, in five space. And so we can, two spheres link. And so, so this is now, if you think of this as sort of a, anyway, these barbell maps, there's just higher dimensional versions of them. And so, right. So anyway, conjecture of this, this barbell map of this, this thing gives a non-trivial diff of S one cross B four. And anyway, I'm out of time, but I'll just show you the slide anyway. And, and this is another example of a bad thing becoming maybe the good thing. The bad thing here was that, well, all these barbell diff, all these barbell maps become isotopically standard, well, the least ones that we constructed all become isotopically, well, anyone. They all become isotopically standard modulo diff before when you lift up high enough covering space. So from the point of view of, of constructing knotted three ball, three spheres in four space, well, they, these things, you're not gonna get examples this way. On the other hand, because of this, things becoming stan isotopically trivial, that gives a potential way of, of constructing diff-humorphisms that are hematopic to the identity and hyperbolic form manifolds that are not isotopic to the identity. So, sorry to go four minutes over time, but anyway, thank you for your attention. Any questions? Any? So, well, Ryan Budney, he thinks these things are also non-trivial in as these diff-humorphisms as they're, he thinks they're topologically not isotopic and there's just things he has to sort out in his proof, but anyway, so in a work in progress that's what, that's what he's trying to do and I think it's probably pretty likely. I mean, it's probably pretty likely too whether he gets it to function, that's another matter. Any more questions? Right, if not, let's thank the speaker again.