 there is a welcome reception today at 7 p.m. at on the terrace of the Adriatico guesthouse everyone please come we are assured that there will be enough food that nobody will be hungry so that's one thing seven o'clock this evening the next thing is a reminder that our schedule tomorrow has changed a bit so here it is on the chalkboard here so the first two have not changed but there's a coffee break after coal and then the coffee break will only be 30 minutes instead of 45 minutes and then we'll have Claudius at 11 15 and then 15 minute break and then Jake at 12 15 and that will be it and then there will be lunch afterwards so hopefully we'll be you know everybody will be able to get lunch I think lunch closes at 145 so we should be fine great third announcement is our next talk is going to be our one remote talk of the conference so what this means is that if you have questions please flag I guess me down or maybe Paul and what we will do is we will run a microphone up to you we have two microphones and the microphones will normally be off but if you have a question we'll turn it on you can ask the question it will interrupt Peter during his talk or if this is afterwards it'll just cut in then and then you can ask your question hopefully this will work okay this is the first and only time we have to try this so we'll see and if you are instead joining us remotely then you can just you know interrupt and ask questions as normal for as we're all unfortunately familiar with with zoom now okay great so our last talk of the day we're very pleased to have Peter Cronheimer telling us about algebraic curves in instanton homology great thanks very much Lenny it's difficult giving a talk by zoom and not having audience feedback so as Lenny said please feel free to shout into up say something nice I'm very glad if you participate that way so I guess amongst the many slices of good fortune that have come my way in my life I want to mention two is relevant to this talk slice number one is the opportunity I had to begin my doctoral work and I think what was a golden age of low-dimensional topology I wouldn't say the golden age because there've been many and there'll be many more and they all one one into the next but the mid-1980s with introduction of gauge theory into topology and Jones polynomial on a different angle was certainly a great time to be writing the PhD the second slice of good fortune after my doctoral work as a postdoc my second slice of good fortune was to had the opportunity to be part of Tom's mathematical journey and a collaboration which has carried on now for I'm not quite sure how many years and no collaboration here this is the Institute for advanced study but I arrived there in the 1987 as a postdoc also there was Gordon Martich and through Gordon and her partner Mugenco I first met Tom I think that at dinner at their place Tom was passing through Gordon Mugenco lived on the other side of von Neumann Drive from where I had in-state housing Gordon I told me about this guy Tom and the work he'd done in the spirit of cliff top stuff developing gluing and obstruction theory for young mills equations on manifolds with cynical ends that work of Tom's in collaboration with Bob Gompf led to the first examples of irreducible simply connected for manifolds that were not algebraic surfaces the first calculation of any dancing in variants that were not of algebraic surfaces and later in collaboration with John Morgan and Danny Rubin Tom's work led to the first obstructions to embedding surfaces in for manifolds first new obstructions beyond the case of spheres that was obstructions to embed in Torai in for manifolds genius one I spent the summer of 1988 and 89 MSRI as it was then called in in Berkeley and that's my first really talk math with Tom I think in a serious way and beyond the genius one case Tom and I talked about possibility of obstructions to embedding surfaces of high genus in for manifolds you should keep an eye interestingly on the Google review ratings for the different mathematical institutes that come up in this slide I left MSRI in the summer of 1989 I guess leaving my car with Tom to try and sell and after a few emails I got back together with Tom for more long working in Oberwollfach and it was here at Oberwoll we spent a whole month together working and at the end of that month I think we had most of the proof of the local Tom conjecture and take the example Milner's conjecture about the unnotting number of Taurus knots the project was actually finished up shortly after we left Oberwollfach after that Tom was at Caltech for a while and I was at Merton College Oxford we visited we also spent time in New York and what we were exploring then in the early 90s was really an extension of what we did in Oberwollfach but it kind of took a new took a new flavor in Caltech we'd realized that the work we'd done at Oberwollfach should lead slightly more generally to constraints on dolls and polynomial invariance of four manifolds relations satisfied by them and tied up with that should be further constraints on the genus of embedded surfaces in four manifolds and eventually all this just sort of fell into place rather beautifully and it became this paper about 150 pages introducing basic classes of four manifolds the adjunct inequality which constrains the genus of embedded surfaces all forced four manifolds satisfying a simple type condition this I think I mean this is still the paper that I've written with Tom which still gives me goosebumps I think if I look at that paper now I think that was some serious effort that I haven't ever quite reproduced in collaboration with Tom I think that's probably the longest paper we also wrote to so it's that work which is really behind the developments I'm going to talk about today which we can somehow see as an extension of what we did back then what were those mathematical themes back in the 90s in that paper as I said relations satisfied by Dawson's polynomial invariance so polynomial I'm quoting I just called him capital Q here names vary x is some smooth four manifold and alpha one to alpha D are chosen two-dimensional homology classes if you like in in X and to such things Jonathan associates a rational number by sort of counting instantons at work a viewpoint that was sort of present in our work in the 90s only sort of partially but is perhaps the more modern way to think about these things is distinct not directly about the invariance of four on a fold but the related flow homology of three manifolds the incident on for homology and to think not about the dance invariance but the operators on for homology so here why is a three manifold alphas a two-dimensional class in why and alpha defines a endomorphism of the instanton for homology and we can ask for what relations this operator satisfies in particular what's its characteristic polynomial whether it's eigenvalues this was a question which was answered in important fundamental case by J. Munoz and in the mid 90s for the case of the circle times a ribbon surface this was with the basic case Munoz completely identified the ring structure of the incident on homology of this three manifold in particular identifiers operator alpha and it's and its properties what I'm talking about today is not directly the incident on homology of three manifolds as it's such but singular incident on homology single incident homology associates to a pair y three manifold and K a link associates to it an obedient group I I have yk incident on homology admissible means that the link K has odd winding number in the three manifold with respect to some integer integer class I'm also going to be talking about incident homology with local coefficients the straight-up incident homology might have Z or rational coefficients local coefficients on the configuration space makes a slightly more sophisticated gadget gamma this in the notation here stands for the local coefficient system and this is a no longer just an obedient group it's a a module over the ring of long series in some variable which I'll call tau finite normal series I should make so as I mentioned a couple of slides back given a two-dimensional homology class now in this singular case I want to think about y and K perhaps a defining orbital technically I might think of alpha as defining something like pairing with two-dimensional orbital chronology classes but these two-dimensional classes now define still operators on the singular incident on homology of a pair y and K there's another sort of operator in the singular homology case which is important to this talk which is associated to a point on the link so given a point P really just a choice of component of the link there's an operator delta also acting on the incident on homology both of these sorts of operators exist also in the case of local coefficients gamma so the incident homology with local coefficients gamma I construction is a module over the ring of the long series in town but as in addition these commuting operators alpha for the two dimensional multi-class chosen there and an operator delta one for each component of the link there's something which in general doesn't make a whole lot of good sense but does make good sense in the cases we're going to talk about which is to consider incident homology module of the constraint that all the deltas are equal set them all equal which doesn't do anything if K is already or not if there's only one delta but if it's a link just for the sake of this talk and nowhere else I'll talk about this as the restricted incident on homology it's going to be a long series module and it's got now a single operator alpha and a single operator delta which was any base point on on the on the link so where does single incident homology come from in its definition I won't say very much about it at all but it begins with the study of the representation variety associated the pair y and k these are the homomorphisms of their conjugation homomorphisms from the fundamental group of like flat connections and the fundamental group of the complement of the link to in this case the group s u2 satisfy this condition which is sometimes called traceless it's the condition that the meridians that that red curve over there map to things in the kind of goosey class of this element of s u2 that's the kind of goosey class of traceless elements of s u2 so that's a nice representation body associated to the pair in this talk there are two important and closely related examples about the simplest examples you can get here z n is and the three manifold is just s1 times s2 the link k just consists of n parallel strands running in the s1 direction n is odd for the admissibility condition y n is pretty similar except that what's inside is not a link with any components it's a not because as the strands go up they rotate one click and spend most of my time talking about why and in this talk but really secretly z n which is some of the fundamental object here and where most of the math really happens I mentioned the representation by the pair what is the representation by in the case of z n it consists of two pieces each of which is a copy of something just familiar in other contexts it's the two copies of the representation body of the cross section here which is the sphere with n marked points if you like an orbital sphere that representation by the overall sphere is a described as a modular space of certain parabolic bundles and it's a very well studied final variety the representation body of z n is then two copies of that so what is instant homology in this particular context is the z n guy here you can think of it starting with the ordinary chronology of the representation body and then in some sense deforming it the deformation includes adding instant on terms to everything in sight everything inside in this case when you change from the chronology to the incident homology actually the underlying obedient group doesn't change there are no actual differentials for supermodel the complex changing the differential nothing happens but what does change is the operators so you should really think of this as a defamation of the operators alpha and delta to obtain the instant homology from the ordinary chronology sorry here we go so the chronology of that representation body or the corresponding representation body for the orbital sphere and has a long history first three names there are in purple because they were studying really not this orbital sphere but the smooth surface of genus G in the context of gaseous seminal work there was by T and bought some of the first results about the structure of the chronology ring were obtained by Thaddeus and by Cohen there are many other names that could be included there and in the context of this orbital to sphere something like a tear what result was obtained by Hans Bowden and the car Morgie ring was studied by Johnson Weitzman and most relevantly to this talk by Ethan Street in his 2012 PhD a theme that appears prominently in some of this in particular the Earl Curran paper is the description of the car Morgie as presented by generators and relations that the generators are the tier bot generators but the relations sometimes talked about as the month relations coming from index theory and algebraic geometry the instant on homology in the case of a smooth surface of genus G as I mentioned before was analyzed successfully by Munoz Ethan Street's thesis analyzed this orbital case this case of ZN exactly so the insormology of ZN was entirely determined by by street at generalizations and a partial calculation of the ring was obtained by Yixi and Bo Yu Zhang for the case of an orbital of higher genus in this talk I want to describe a defamation you ought to think of local coefficients as just another defamation of the instant on homology further defamation again you shouldn't really think of the underlying module the group is changing really except extend the coefficients to this ring of long series what changes now is the operators so that's really what we want to understand how these operators deform and how do they have to stick polynomials change that's the case of ZN the story for YN is what I'm going to show in the next few slides you should think of it as being essentially the same the beginning point isn't quite the carmology ring of ZN anymore it's not really the carmology ring of instant homology of representation by the YN in our context the best way to think about it is just to take the ordinary carmology of that representation body and do this restricted thing just kind of set all the deltas equal that becomes something rather simple instant homology could be entirely described using Eastern streets thesis and what I want to talk about today is the defamation with local coefficients so what is this gadget over here it's some module for a ring as I said in the last series in town as local coefficients with two interesting commuting operators alpha and delta and my court wants to answer the questions to what relations are satisfied by alpha and delta in particular as they're commuting operators they have simultaneous eigenvalues what are those eigenvalues for alpha and delta as a pair so because we're working over the losses I could think of towers just a real or complex parameter if I want in which case as I vary tau I've just been someone was just some some complex vector space of fixed dimension alpha and delta just matrices of operators on that finite dimensional vector space commuting but depending on towers a parameter and I can ask what the eigenvalues of alpha and delta are as functions of tau just a different way of looking at the same thing so next several sides are going to show you the answer. Beginning with a case which is sort of trivial but still somehow non trivial and you do these pictures the instant homology of why and that's three strands running up s one times s two few strands and just with one click of rotation so it's a not in s one times s two with winding numbers three what is the homology of the representation body that the representation body is just two points the operator alpha and delta about zero on the on the homology and a simultaneous eigenvalues of alpha and delta this graph here this is the alpha axis I hope you can see the alpha label there and the delta label there this point here is the simultaneous eigenvalues of alpha and delta but thinking of this as a module I can think of this as a sheaf on the plane in which case it's two copies of the skyscraper sheaf located at zero zero and that's the homology of representation. According to eastern streets calculations what happens when you take the instant on homology that's a defamation of this picture. Those two copies of skyscraper sheaf they just kind of move and there they are now with eigenvalues of alpha being one and minus one and the idea of delta still being zero so two skyscraper sheafs at those two points already in this very simple case it's quite interesting to ask what happens in the local coefficient system which you vary tau. The answer is that those eigenvalues move in the plane and they move along these curves. That's actually some smooth rational quartet curve in the plane and that's a very tau and those green dots what that's what happens when tau is more point six. Tau equals one is trivial coefficients that's where the pink dots are and as you vary tau away from one the eigenvalues move along these algebraic curves. Let's draw the picture it gets a bit more interesting for n equals five five strands. Again the eigenvalues of alpha and delta just for the ordinary carmology are just zero. So everything's at the origin there. There's two copies of that blob at the origin but that blob is no longer a skyscraper sheaf but got two copies of what two copies of the fat point at zero zero fat point first infestable neighborhood of the origin. It's a sub scheme of the plane of length three two copies of that direction six sitting there at the origin. That's just the ordinary carmology ring and there's sort of restricted versions setting five deltas that will equal. Streets thesis tells us what happens when we deform to the instant carmology and this is what happens the eigenvalues of alpha now become the odd integers three one minus one and minus three. At three and minus three there's just a skyscraper sheaf here. The eigenspace at this point it is two dimensional. It's not two copies of the skyscraper sheaf. It's a sub scheme of the plane of length two. It's two points of coalesced in the direction of the delta axis there. The total length here is six one plus two plus two plus one. So that blob of size six has just become these six guys. That's a streets thesis. What happens when we introduce local coefficients is that these eigenvalues move along now a more interesting algebraic curve. That's what happens when tau I think is 0.7 in this picture. The six blobs will become six distinct eigenvalue pairs in the plane and equals seven. Ordinary carmology is two copies of the second interest neighborhood at zero zero. This is how it breaks up just according to street eigenvalues now all integers in the instant on homology. These are sub schemes of the plane of length one, two, three and the eigenvalues of alpha plus or minus one, three and five. What's happening here is a little bit more interesting. It's not three points having coalesced parallel to the delta axis. It's some curvil India sub scheme of the plane. Three points of actually sort of coalesced along the parabola at that point locally. This is what happens when we introduce local coefficients. Again, the eigenvalues move along is interesting algebraic curves. So that that blue curve there, that algebraic curve, that's the curve of relation just describing the relation between alpha and delta some polynomial in alpha and delta is zero. What is that polynomial in alpha and delta? This is a small zoom in of the rather large mathematical output, which describes what that polynomial is. It's got degree 24 now from delta and it looks a mess. I think any polynomial degree 24 is likely to be a mess. It's just an irreducible polynomial. What's it gonna look like? Well, it's gonna look something like this. But the calculation of this thing is kind of what I want to explain a bit for the main part of this talk. Very briefly, this is what happens in n equals nine. It looks kind of similar. eigenvalues of alpha now go from seven down to minus seven. Here's the case n equals 11. The equation of this blue curve is now truly horrendous. It's got degree 60. That number there is the coefficient of alpha to 51 times delta to the nine. Again, it's some irreducible polynomial of degree 60 in two variables, alpha and delta. As an equation of alpha delta, it's sort of quite scary. But there's actually something which is, you should think of tau. Your tau is a coordinate on this on this curve. It's a function, actually a two valued function of this blue curve, describing how the eigenvalue point is green points move. You should really think of the curve in three space where the third coordinate is tau. And that blue curve is the projection to the alpha delta plane for getting the tau coordinate. If you take a space curve and project it to the plane, the equation gets a whole lot more complicated. And that's kind of somehow why the polynomial alpha delta is so horrific. This is the next case up n equals 13, a partial picture of the curve. But now in three space, the coordinates are slightly different. But basically, this is in three space with coordinates tau, delta and alpha describing this algebraic curve. So if n equals 13, this is the object we kind of really want to study this instant on homology with local coefficients. And what it is, to say things another way, it's just a ring of regular functions on this algebraic curve in three space. So to describe this instant noncourmology, we need to understand what is this curve, what are its equations, what characterizes it. So how these equations computed in principle, the creation of these curves, they define using instant on homology. So the answer depends on calculating some instantones, which seems impractical. But essentially, as in monosis calculation, or the second version of monosis calculation, we can eventually obtain the answer only by understanding the simplest, non flat instance of the instance of smallest non zero action. So how does this happen? As we vary tau, we just get some bunch of points in the plane, the alpha delta plane. Here, in the case n equals 11, you actually get 15 points in the alpha delta plane, I want to describe what those points are. I'm only describing a point in the Hilbert scheme, Hilbert scheme of 15 tuples of points in in C2. The ordinary chronology, here's just a blob at the origin, I can also think of that as an element of the Hilbert scheme. So really, what I want to describe these 15 points is some coordinates on the Hilbert scheme, which should describe where these points are. And they start off at the interest of a neighborhood or bundle together here. And here they're deformed as we vary the deformation, there's a combination of the instant on deformation and then the local coefficient of deformation. So what affine coordinates on some Zariski open neighborhood of this point in the Hilbert scheme. And there's a neat way to do this with determinant variety. So s here is as relevant as the case of n equals 11. s is a matrix of here was just variables s i j. It's got dimensions five by six. The simplest sort of thing I can consider in this case, and call a determinant variety is the locus of such matrices which do not have full rank, the general rank would be five, but let's consider the variety of such matrices s where the rank is four or less. That turns out to be some co-dimension to locus in the space of matrices s. I now want to do something relevant to the plane. Let's consider that the entries of this matrix are in homogeneous linear functions of degree one in some variable uppercase x and uppercase y with some coefficients, which I'm just going to think of the coefficients ABC is fixed. And think of varying x y as coordinates of a point in the plane. And as I vary the point x y in the plane, I can ask, when is this matrix not a full length? Which values of x and y? Again, that's co-dimension two. So it's going to be sort of a finite set of points, capital X, capital Y in the plane. Not having full rank is the same as saying that the minors of this matrix, the five by five minors are all zero. So it's defined by these polynomial conditions. As such, I can really think of this configuration of 15 points x y is that's more generally under the Hilbert scheme of 15 tuples of points. Sub schemes of the plane of length 15. So that's the problem there in that box. Given thinking ABC is fixed, find the points x and y in the plane say C2, such that the rank is less than five. And as I said, it's a group of vanishing of some polynomials. Polynomials would become polynomials in x and y. I said I want to think of ABC as fixed, but here I'm taking a very slightly different point of view. Let's let's think about s as first of all s one, which is the degree one part in x and y, and then s zero, which is the constant part in x and y. Let's think about A and B now is fixed. And let's vary the coefficients C. So if each choice of A, B and C, I get some configuration of points in the plane, say 15 points. And as I vary C, these points will will move around in the plane. The miners of s one itself, when all the C's are zero, that turns out just to be is for generic choice of A and B is that that's just the maximal ideal fifth power. That's just the blob at the origin, fourth in decimal neighborhood of zero. I add these zero thought of terms that the Cij and I got a general element and some open neighborhood in the Hilbert scheme. This is here it is pictorial. This is the vanishing of the miners of the homogeneous matrix. This is the vanishing of the miners of s one plus s zero. And as I vary s zero, these points will will move around. So as it turns out, there's a very simple way to describe what's going on for y 11, for example, or any any in exactly these terms. So this is really a special instance of the general story as talked about before. This large piece here with alphas and deltas think of that as s one. Capital Delta is actually just lowercase delta divided by two just to make the type setting a little bit more compact. This is a matrix of linear forms in two variables. They used to be x and y and now they just alpha and delta. This is a term which is zero thought of this is the s zero, zero of order in alpha and delta. It's a bunch of scalars in the coefficient ring law series in town. These matrices may look a bit scary, but you can kind of see the pattern. This is a two band diagonal matrix. The coefficients here 1915 117, they does move in a simple geometric series. I guess arithmetic progression, not geometric. The same with these coefficients. Here there's some weights tower at the front, but this is a three band sort of constant diagonal matrix. Again, the coefficients just move a little arithmetic progression here. These matrices aren't very hard to write down. And they just completely describe what's going on for the instant on curmology with local coefficients. I've got this tower dependent as zero. Locals where s does not have full rank will be some point bunch of points in the alpha delta plane. As I vary tower, these points will move around exactly as those green points move around moving along those blue curves in the previous pictures. Another way to say that is that this instant on curmology ring with local coefficients is to ring functions again on some curve in three space. It's the curve defined by the vanishing of the five by five miners of that five by six matrix with coordinates tau, alpha and delta. I want to just pause to emphasize how surprising this result is. In Minot's work and in Ethan Street's work, the instant on curmology of these operators alpha and delta completely described for closely related situations to this. But they're only described by some recursive formula to compute these relations. There's some recursion that throws these things out one by one. This is a very compact and closed form for some interest essentially a generalization of Ethan Street's calculation. Where do these matrices actually come from? What are the ingredients of this calculation? I don't have much time to talk about this, but the matrix S1 of linear forms that really involves the ordinary curmology of the representation variety of Zn. And what it divides from is an explicit presentation of the ordinary curmology. The curmology is described by these Montford relations. But in this particular case of the orbital sphere, there's an explicit product formula of these Montford relations. I call these the Rufka relations for the orbital sphere. As an explicit product of linear forms, it's from that that you can write down very easily the matrix S1 of essentially with scissor genes for these relations in the ordinary curmology. As I said earlier, this is really parallel to work of Minot's matrix S0. That's the instant on deformation and the local coefficient deformation. By a small miracle of this algebraic geometry and these coordinates on the Hilbert scheme, you can completely understand S0 only if you understand just the next lowest term in the relations. And that just depends on the instant non-modulated spaces of smallest non-zero action. So those were just the ingredients of the calculation which led to this very explicit form of the description of the instant on curmology. I should say a description of my collaboration with Tom that this kind of took a long time. I mean, this was pandemic work, and it was kind of months of effort for the case of n equals five, which we really studied very explicitly using a description of the representation variety as CP2 blown up at five points, understanding explicit instantons by understanding rational curves in that far amount of thought. Big mathematical calculation with large matrices and finally finding the characteristic polynomials of alpha and delta. To go further, and when we done n equals five, I think I was ready to give up. Tom said we should just, you know, we done more than that in the nineties with that 150 page paper. We ought to be able to do more in 2020-21. Eventually we found for general and a description of the operators alpha and delta as functions of tau, but only via a rather elaborate recursive calculation sort of similar to Ethan's recursive calculation in some ways and that of Minos in his first paper on s one times sigma g. It was, I think it surprised the both of us, the final answer could actually be written down in such a small closed form by these scissor descriptions of the coordinates on Hilbert's scheme. There were intended applications for this along the way that we haven't returned to. Having done the case n equals five a long time ago. Our motivation was to describe the sprained singlet incident cohomology of torus knots, in particular, five n torus knots, starting with this z n and y n and applying some surgery triangles. That's something we succeeded in doing for t five four and t five six. It's something we haven't yet returned to for a higher n, but that's certainly the intended application originally. I think at this point, the intent application is just to look at the beautiful, the beautiful answers. So I'm happy to answer any questions, but that's the end of my talk for now. And thank you very much for listening. And again, I'm sorry, not there in person, but I very much enjoyed being there. I presume. Thank you. Thanks, Peter. Are there questions for Peter? Don't give him a bad impression of our people here. Jake. Sorry, this is a dumb question. But can you remind me what the framed in which one is the framed instant on homology? Yes. So dance actually isn't that simple. But when we first started doing this, the famed is not homology, but for us was where you take this not y k. And you connect some with a theta graph in the in the in the three sphere. And the local coefficient system should involve three variables, tau one, tau two, tau three, attached to the three arcs of the theta graph and maybe a variable tau associated on took to the not in y. That the problem with that guy there, having a graph as we don't really know how to do that except with them. Z more two coefficients are originally homology calculations all with Z more two coefficients, which was very scary, indeed. An alternative is to take and connect some not with a theta graph, but with essentially the Z three guy three strands in in s one times s two, whichever way you look at it. Perhaps the more fundamental viewpoint is in the work of Addie Damian and Chris Scoludo. So you could think of these calculations as relevant to a calculation, what they might call the the equivalent version of single it's the homology for for not in in s three, for example, such as the torsion. Peter. So what what not do you know this homology for now? I'm not sure what the complete list is. I'm reading. We know it for the three n torsion knots. We know it for the five four and I think the five six torsion knots. We know it for the not 74. I don't mean the torsion not 74. I mean the not 74 in the not tables. And a few other small knots, which you can make. I think the twist knots at some point. The motivation originally came from the hope that we could use this, for example, for the not 74 to detect the difference between clasp number, slice genus for connected sums of many copies of a knot. It turns out we couldn't do that in the way we wanted it. So, you know, you had to and Zempke found a difference for the seven four knot. And the sorts of results they obtained using group so long, we could sort of reproduce using the single it's not homology, similar to the loop stone calculation. What we weren't able to do was what Damien's Gerudo were able to do, which is distinguish the slice genus from the positive clasp number for the seven. That's what something wanted to do for the seven four knot. And the, I think the motivation is still to try and compute some more interesting knots and see whether anything more interesting starts to happen in this single incident case. The double points, introducing double points in some embedded surface or corresponds to multiplication by one element of this coefficient ring and adding genus corresponds to multiplication by some other. And if the module you're looking at is some interesting thing, like a sheaf on some algebraic curve of large genus, then multiplication by two different elements is something where there's interesting relations which are sort of go beyond just integer orders of vanishing at a single point for some. But I think that that's still the motivation in part for the calculation, but just a small knot so far. Another question. Is there an explicit description of the matrix for any odd n? So here, I guess, is for n equals to 11. And how about? Oh, yeah, I mean, it's, um, it's, you know, said to the same thing. And so as you, if you write n, which in this case is 11 as two K plus one, so K is five. There's just some matrix s. Instead of being five by six, it's K by K plus one. It's still two bands. These coefficients just sort of follow the same little arithmetic progression in steps of four just continues a bit longer and starts at a slightly different place. So if you gaze at this matrix long enough, you ought to be able to guess what it looks like for n equals 15 or 19 or, or 101. It's, you just kind of continue the pattern and make the matrices just, just bigger. Is that? Yeah. So does this mean for any n? So the eye of why gamma is already explicitly computed? It's explicitly computed in the sense that it's the locus where the rank of s is, is not full where s is a very explicit matrix. It's given explicitly by the banishing of the minors. The m by m minus was certain m plus one by m matrix. Yeah. Thank you. Other questions? Tom, you want to say something? I won't ask any questions because I probably know what he's talking about. But of course, unfortunately, since Peter's not here, I don't have a chance to thank him in, you know, in person in front of other people. But I think this is as good as close as it gets. It's been an amazing ride. And I'm amazed that it still keeps going and it's fun. And it's really, I mean, you can't get luckier than having a collaborator like Peter. So that's all I want to say. Thanks. Anyone want to top that? Okay, if not, let's thank Peter again.