 It's a tremendous honor to be asked to speak at a conference in honor of Marceau Bojet, who was a towering figure of 20th century differential geometry and had so many other contributions in other directions in mathematics. In my own personal development, he was absolutely fundamental as a graduate student studying the sphere theorem. It was maybe the first time I'd really seen how much information you could get out of very carefully estimating things and being absolutely persistent about finding the right way to do it. It made a big impression on me. At the time, I was completely unaware of his thesis work, which would become fundamental for my own mathematical career. His thesis work was in, was of course, classifying the Romanian holonomies. He was he who discovered that G2 and Spin 7 and Dimension 7 and 8 were possible as holonomy groups. And although they didn't actually construct any examples, that would not happen for another nearly 30 years. That amazing attention to detail and care that went into actually doing that classification was, made a huge impression on me. And it's probably, you may have noticed and if you look in some of Berger's early papers at that time, there are big lists of tables of things, you know, the answers that you get after you do a very, very careful and detailed analysis of the possibilities. And of course, that made a big impression on me too. And so, one of the things I want to talk about today is an outgrowth of this special holonomy, this work on special holonomy manifolds, looking at, again, looking at geometrically natural conditions on special holonomy manifolds and trying to, and trying to determine when that can be used for, for constructing new examples. One thing that was always very mysterious when we first came, produced proofs that these, that these special holonomy metrics existed in Dimension 7 and 8 was that the equations themselves seemed to be very complicated and trying to understand locally, which was, which was how I got involved in the story to begin with, was, just didn't seem to be related to any, any well-known or, or understandable system of PDE. Finding, finding a way to construct examples was, was hard. And Simon and I worked on the Co-Homogeneity-1 case and, which is a geometrically natural condition. And were able to construct some complete examples, the first known complete examples of these special holonomy metrics. But the more you study it and the more you look at it, the more you realize the intuition that there should be some connection with integrable systems. It's not exactly clear what that connection should be, even now, what that, but, but finding ways to look at either reductions or special cases geometric by adding geometrically natural conditions was the thing that is the theme that I want to take up because it does in fact lead to connections with integrable systems. And, and that's really the message I want to talk about today is that we have a way of constructing solutions now, more ways of constructing solutions by looking at things as, as integrable systems. But finding that reduction to integrable systems takes, takes, takes a couple of new ideas. And that's what I want to talk about. Just let me remind you of basically what the, what the story is for a Riemannian manifold with a holonomy group H, there it is, holonomy group H sitting in the orthogonal group with Li-algebra fractur H in SON. The, and this is very classical. It goes, goes right back to, right back to Berger's original studies. The, the, what one can do is of course write down the structure equations on the bundle of, you know, the, on the holonomy bundle of associated to the, associated to this manifold that is pick a frame, pick a, pick a particular frame at one point and translate it around by parallel transport and you'll get a, you'll get a sub bundle of the frame bundle, the, that will be actually an H bundle, H reduction of the, of the orthogonal frame bundle. And on that bundle you'll have the structure, the standard structure, the carton structure equations. The differential of the so-called soldering form is minus the connection form which, the soldering form and then the curvature, the expression of the curvature as a, as an operator as, as a linear map from the, as a linear map from actually it has in fact from the, from the algebra to itself the space of curvature tensors actually takes values in the, in the quadratic. Sim 2 of the, of the, of the holonomy algebra in fact it's the kernel of the natural map of the thinking of, thinking of the holonomy algebra as sitting in S O N which is lambda 2 of R N just taking the, just sending it in by wedge product into lambda 4. The kernel is in fact the set of things that, that satisfy the Bianchi identities to be the curvature tensor of a, of a Riemannian manifold with holonmy H. Now in addition to the, in addition to the, the first Bianchi identity which is what gives us that the curvature tensor takes values in this particular algebraically defined subspace. The second Bianchi identity tells you that, tells you that when you differentiate the curvature the, the first covariant derivative takes values in a certain subspace of the, of all the you know maps of, of Varian into, into the space of curvature tensors and in fact it was Berger's, Berger's careful analysis of these two spaces that led to his, led to his classification and in, in particular the case where this guy where H acts irreducibly and, and, and this space is more than one dimensional and this space is, is in some sense large enough to, to generate the entire, the, the variations of the curvature that show up here. Those are the, those are the two conditions, the, the Berger conditions and, and in fact it turned out that, that that, that those conditions which Berger proved were necessary ultimately after a long, many years later we were able to show that they were sufficient and, and so his, his criteria really were at the heart of the matter even though it took us a long time to realize that that was the, that was the essential thing. I'll, I'll just give a, a brief example, the concrete example that everybody can see. If you look at issue two sitting in, sitting in SO4, the, there's the, the connection form of the, on the, on the issue two bundle takes values in the issue two, the algebra, the structure equations. When you, when you compute the curvature you get that the, you get that the curvature takes values. This is, these are the two forms that correspond to the Lie algebra itself. It's actually a symmetric map from the two, from the Lie algebra to itself. And it's required, it turns out it has to have trace zero. That's the, that's the, the Bianchi identity comes in. So what that says is that the, is that the first that the space of curvature tensors is in fact traceless symmetric three by three matrices is irreducible representation R5. And computation shows that the first covariant derivatives of curvature are this irreducible representation of issue two, which is simply a sem five of C2. It's a 12 dimensional real space. And, and in fact if you, as I'll say a little bit later, in a certain sense that those algebraic facts are all you need to, to apply Carton's machinery to prove that, prove that issue two occurs as holonomie of, of, of four manifolds. This was of course long before Kaler geometry or anybody knew anything about Kalabiow or anything like that. In fact this was the first known non-trivial case of an irreducible holonomie. Carton just mentions it as a fact in his, in his little book on Riemannian geometry. He doesn't give a proof or, or, or or any argument he just says, he just says the, the metrics with this holonomie depend on two functions of three variables. We'll, I'll come back to where this two and three come from in a little bit. So anyway the basic pro, the basic problem is once you, once you believe that there exists such, such things with a given holonomie is how to classify what the solutions are. That is determine up to diffumorphism how many, in some sense how many solutions there are to these equations. Where theta takes values in the Lie algebra H and R takes values in the, in the space of curvature tensors. This is a classic kind of problem that shows up, shows up in Carton a lot. And, and there's a, there's a well-developed machinery for, for understanding it. What I'm going to do is, as I'm going to consider a, a modified problem is, is instead of allowing the curvature tensor to take values in all of the, all of the possible curvature space, I want to look at a natural subspace of the, of the space of curvature tensors and ask what are the solutions who take that, that take values in, where the curvature takes values in some subspace. And in order for this to be a geometrically natural condition, what I, what I require is that, is that I take this, this subset of the set of possible curvature tensors and I require it to be invariant under the action of H. So it's some sub, you know, some H invariant sub, hopefully sub manifold but not necessarily. It could be a variety with singularities sitting inside, sitting inside the space of all curvature tensors. And we look for the solutions whose curvature tensors take values in that algebraic subset. I'll illustrate this in a minute. So for example, what would, what would happen in the case of SU2 as holonomie sitting in S4? Remember that we, we remarked that the space of curvature tensors is in fact the traceless symmetric 3 by 3 matrices being acted on by R by SO3. And, and of course the invariance of this, of this quantity are in fact the symmetric functions of the eigenvalues of this traceless symmetric matrix. And of course the traceless 0, that would be sigma 1. So it's just sigma 2 and sigma 3. The sigma 3 being the determinant and sigma 2 being the sum of the squares of the eigenvalues. Well, sorry, the sum of the, the symmetric function of the, of the eigenvalues. And, and of course because it's a real symmetric matrix, there's an inequality between the, between these two functions. But if you look inside this locus where the inequality holds, picking an invariant subset in the plane in the, in the locus where this inequality holds will define a subset of these matrices whose sigma 2 and sigma 3 live in this, live in this region. Just to, before looking at that in detail, let me just mention that what happens in the, in the various special holonomy cases that most famous and most thoroughly studied is the case of, the Kalabi-Al case of SUN, seeing SO2 in, there the space of curvature tensors is again irreducible. It's the, it's the quartic polynomials on CEN that are, that are by degree 2, 2 and, and that are harmonic. That's the space of curvature tensors. And, and the, and in particular in the cases n equals 2 and n equals 3, which I'll say more about, these show up, these show up in, in the study of, of course, you know, Kalabi-Al surfaces and in connections with strain theory in nearly Kaler geometry. The thing that got me interested in this at the time, you know, more than about 30, about what, 30, well, it's hard to believe, 35 years ago, that was this open question at that time about whether or not G2 actually could possibly occur as holonomy. It was known but from the work of Alexievsky and, and Bonin, that the space of curvature tensors was, in this case, was an irreducible representation of G2, V2, V02 in terms of the highest weights, which is a 7 new, 7 dimensional space. And, and the, and spin 7 in SO8, its space of curvature tensors turns out to be, turns out to be V020 for SO7 representation, which has space of dimension 168. I'll come back to these numbers a little later. Sorry. Yeah. So let's go back to the, to the, I want to illustrate what happens when you, when you do impose such a condition in the, in these issue two holonomy case. First, without any restriction, the, when you compute the characters of this, of the tableau that's associated with this, with this, with this curvature and the first prolongation of curvature, compute the characters. The characters are, are given as follows. S1, S2, S3, 4567 is, is the last nonzero when S3 is 2. And, and because this dimension of A1 is in fact, according to Carton's count, 2 times S2 plus 3 times S3, that this satisfies Carton's conditions for, for inviolativity, so it's inviolative and that's basically Carton's proof. I, I believe would be Carton's proof. He, like I said, he never actually, as far as we can say, as far as we know, wrote down any explicit argument, but this would be the natural argument that he would have made if he had. So it's inviolative and the fact that it's S3 is, S3 is 2 is the last nonzero character. That's the statement that the general solution depends on two functions of three variables, up to diffeomorphism. But when we go to the algebraically special case, the systems we need to study are not always inviolative in fact. And, and turns out you have to prolong the structure equations. I'm going to give a particular example, the case where the, the case where the curvature tensor has a double root. In which case, if it, if it has a, if it has a double root, you can always, you can always locally, as long as the root's not zero, you can smoothly diagonalize the, the front, the curvature tensor. And because the trace is zero, the, the double root that forces that to be a minus two. And, and the R cubed there just makes it, turns out to make the calculations easier. Whoops. Okay, yeah. If you differentiate these equations, what I want to illustrate here is if you, if you now take these equations and take their, take their exterior derivatives, what you find is that, well R of course, it's differential, it's a function down on the, down on the four manifold. And so it's differential has a, is, if it's not zero, you can write four R times, times some one form down on the manifold, this U naught, eight and odd up to U three, eight of three. And it turns out that when you differentiate these structure equations, because you've made a frame adaptation, you've actually reduced it to a, reduced it to a circle action. And theta two and theta three, they're the, they're the, I mean the circle action is the theta one there. Rotating that one way in the eight of zero, eight of one. And rotating the other way in the eight of two, eight of three. And, but these guys are no longer part of the, part of the frame bundle when you, when you diagonalize the curvature tensor. And in fact it turns out that differentiating these equations forces theta two and theta three to be these particular combinations for the same use as dr. So what we're saying, what, you know, informally what you would say is that, is that the, the set of one jets of solutions, so, so, well the, the, the two jets of algebraically special solutions have only one invariant, the R. And the three jets will have, will have four more invariants, the U naught through U three. Unfortunately these equations, if you now plug this all in and, and calculate the carton characters, this system's not in volume. And, and as a result, and as a result if you want to know whether or not there are any solutions you have to, you have to work more, you have to differentiate these equations and apply these structure equations and see what happens. What actually happens is this, the, when you differentiate the, those structure equations, the differentials of the use, the, the new unknowns turn out to be expressible in terms of the co-frame. In terms of the known quantities plus it turns out there's a three-parameter family of, of solutions in terms of the derivatives. There's still three new variables, three new invariants that, that are not, that cannot be, that are not functions of the previous known things. And it's still not in volume. You differentiate again and, whoops, I'm having trouble controlling this thing. Differentiate again and, and finally you get to the point where, where the, the derivatives of the new things actually turn out to be expressible as, in fact polynomials in the, in all the things we have so far. So you get, so here's the full list. There are, there are basically eight invariant functions, eight, eight invariant coefficients. And their derivatives are expressed purely in terms of them. And lo and behold, if you try looking for new relations among them by taking exterior derivatives of these equations, you don't get anything that, that is, as Carton would have said, d squared equals zero is an identity in this case. And now Carton's sort of generalization of Lee's third theorem says that there always will exist solutions for any given values of these eight quantities. There'll always exist a, a, a solution to the structure equations when she specified these eight quantities at a point. So what you can say is that, is that solutions do exist, that is, there are Calabi-Almetrix for which the, the, the in dimension four for which the curvature tensor has a repeated root everywhere, but there's only a finite dimensional space of them. They, they exist but they're, but they're, you know, they don't depend on arbitrary functions, they depend on only constants. For comparison, let me go back again to the classical Holonami case with no, no curvature restrictions. That is, if you look, if you look at the, at the SUM case, the Calabi-Al case, the last non-zero character is 2n minus 1 is 2, which is, which is exactly the, the generality of Calabi-Almetrix in, in updated if you morphism locally. And when you do the calculation for g2, it turns out the last non-zero character is s6 is 6. So the generic g2 metric locally depends on six functions of six variables. And, and I'm not sure I'll have time to talk about the nearly, the nearly-caler or the nearly g2 structure case, I'll, I'll maybe reserve, but basically there's a, there's a, one can, one can specify not torsion-free structures but, but structures that have a single non-zero torsion coefficient. This is the nearly-caler and similarly the nearly g2 case. And, and it turns out the counts are the same for those as for the torsion-free case, that is the, the Hullonomy case. And spin seven, the other one, the other exceptional Hullonomy, the, the last non-zero character is s7 is 12, so they depend on 12 functions of seven variables. Well, let me compare that with the curvature restrictions in the possible curvature restrictions in the, in the SU2 case that we've been discussing. The curvature and variance are generated by the s2 and s3 and satisfying that inequality. One thing you could try to do is simply specify the, the, the symmetric functions, the eigenvalues of some constants. That's one choice you could possibly try and look for solutions like that. It turns out that there are no such solutions. The, except for the, except for the flat case. That is, if you put the structure equations in and, and prolong them, you wind up that the only compatible case is c2 and c3 is zero. You could ask for something like sigma three, sigma three being zero. It turns out that's also not evaluative. And, and if you require that the determinant of the, of the curvature tensor be zero everywhere, then in fact the only solution is that the, that the curvature tensor be identically zeroed. On the other hand, this double eigenvalue case, this is what the, this is what happens right on the edge of this inequality. The, this is in fact the condition for the, the eigenvalues be double. That's a curvature tensor with a non-trivial stabilizer, because the double eigenvalue means that you, means that you have the rotation in that double eigenvalue plane. That has a non-trivial stabilizer. And while it's not evaluative, as we saw, if you prolong, you finally wind up that, up to diffumorphism, there's a two-parameter family of inequivalent solutions. Not all of those are complete, but some are. And, and it constructs a, you know, a special family that, that we had not been aware of in the past. It's not algebraic, by the way, it's not too surprising. The, in general, if you just take any, any H invariant subset and look for the solutions whose, whose curvature tensors take value in that subset, that's probably an intractable problem. If you, just the, the sheer number of calculations that you'd have to do, say in, in the issue three case is, is, in, and I've tried, is pretty, is pretty daunting. But what you can look for is special things. In fact, we still don't know if you go back to this case. If you, if you take the general, you know, if you, oh, it's very efficient. There's no chalk. Oh, there is some chalk, right? So if you, if you look in this, if you look in this locus for, you know, sigma two and sigma three, you know, so this is some inside of a cusp. If you just take any algebraic curve sitting in here and, and say, ask for the solution, ask for the, the, ask for the, the collabial metrics whose curvature takes values in some, in some algebraic curve like that, we, we do not know which, which special homonymy metrics, switch collabial metrics, which of these curves can occur when, and non-trivial. That's a, and, and if you sit down and, and say, well, I'm going to derive the necessary insufficient conditions for that, for that curve to be compatible with the, with the collabial metrics, that appears to be computationally intractable. So far, so far, we had no examples of such curves, but we don't know, we don't know how to characterize them. So it's an interesting problem whether or not, whether or not there's some better idea for, for doing that classification. Anyway, as I was saying, the, the, the general case is probably intractable, but, but it's probably true that we can do a classification of the, of these constraints that are either inviolative or their first prolongation is inviolative. That seems to be, seems to be a computationally manageable thing to do. And, and while, while we don't have by any means the, I do not have by any means the, the classification nailed down, there's a, there's a lot of progress that can be made. And, and I want to tell you a little bit about that. What the most, all the ones that have been found so far, turn out to be cases where the, where the subset that we're looking at, all the things in the subset have non-trivial H stabilizers. That is, you know, of course the, the Lee group H acts on this. This is a representation space of H. And, and so you can look for things that have non-trivial stabilizers in there. And that defines some subset, the, the set that consists of curvatures with non-trivial stabilizers. And while it's not a smooth manifold, it's of course, it can be stratified into smooth pieces, it's algebraic. And, and, and looking in that locus turns out to, in fact, all the cases that I know of where you, where you get to inviolativity or compatibility all lie inside this locus that has non-trivial H stabilizers. And, and the, and my, the, you know, in the spirit of, in the spirit of Berger's, you know, writing down a table of examples of, of table of the classification, I thought I would point out what this looks like in the case of, in sort of the next interesting case. Suppose you, suppose we go, I mean the SU2 case we already know, the only case where you have a non-trivial stabilizer in SU2 is exactly the, the double eigenvalue case. And we, and I just went through that analysis that the double eigenvalue case definitely does exist. And, but if we go up one dimension into, into looking at three-dimensional collabial metrics. And we look for, and we ask that the, and we ask, and we look for the space of curvature tensors that have non-trivial stabilizers. Well, non-trivial stabilizers of course a subgroup of say, let's say positive-dimensional stabilizers. A positive-dimensional stabilizer of, of SU3, fortunately there's not SU3 being ranked two, there are not too many subgroups, closed subgroups of SU3. And you can just look at the, at the subgroups. You can look at the subgroups, you can list the subgroups that have non-trivial stabilizers in, in the space of curvature tensors. For example, in this case, U2, which, if you look at the set of curvature tensors that can occur in a three-dimensional collabial that have, that have U2 as a stabilizer at every point, then, then in fact that's a, that's a one-dimensional space sitting inside the, sitting inside this, this space which has dimension 27. And, and the way to think about this U2 is, of course, it's the subgroup of SU3 that splits, splits off a line. And it's easy to show, in fact this is well known, that there's a one-dimensional family of such things. These are the rotationally invariant collabial, collabial metrics in dimension three. That are not, I mean, well, of course, the flat one is the, is, belongs to the family, but there's a one-parameter family of them that are not flat. And if you look at SU2, then it turns out that, turns out that that still has only a one-dimensional family of, family of solutions, family of invariant curvature tensors. Of course, that splits off the, because you're picking out a determinant that C splits as an R plus R. And again, it's still one constant. There's no more, no more solutions than that. If you look at another subgroup, for example, if you look at SO3 sitting inside, sitting inside SU3, if you look at SO3, that also leaves invariant one, a one-dimensional subgroup of the, of the curvature tensors. And SO3 acts as, as preserving the real and imaginary parts of C3. But it turns out that this never occurs. This, there is no, there is no metric, there is no collabial metric with, where the curvature tensor always preserves the splitting like this. On the other hand, if you now go the, you know, sort of the next, if you look at a maximal torus sitting in SU3, maximal torus of course preserves the splitting of C3 into, into three lines. It turns out there's a three-dimensional family of these, of, of curvature tensors that are invariant under the maximal torus. And the analysis that analogous to what I did for the SU2 case, if you look at the, if you ask that the curvature tensor take values in the conjugates of this guy, then it turns out there's an eight-parameter family of such, of such metrics. Most of them are not complete, but some of them are. But they're, it's fairly, fairly large, fairly large family. If you take now the only other, I mean, I've exhausted the rank two groups and the rank one group that acts that, that's not one-dimensional, the rank one subgroups that are not one-dimensional. And then there are the circles that sit inside T2. If you take the generic circle, that is, you know, remember that in T2, just think of the, think of it as S1 cross S1. And then the, the compact subgroups of that are, are, you know, an S1, whoops, an S1 running in two, you know, with two, two slopes P over Q. The generic one has, which looks like this, where P and Q are not zero and one is this circle diagonal matrices. They have the same stabilizer as T2. And so the answer is the same. Is what? Rational number. Rational number, P over Q, yeah. Otherwise, we don't get a closed subgroup. Once you close it up, you get back to T2. There are two other special, whoops, back up. There are two other special cases, though. If you look at things of slope, of slope zero, so it actually fixes one of the, one of the C's and then just acts as the conjugate action in the other two. It turns out the space of curvature tends to be this five dimensional. And there, and there, the calculations, this thing keeps, the calculations are complicated, but they, but they wind up, it appears that the, it appears that this, that you get an integrable system where the solutions depend on two functions of one variable. The question mark is because there's a, what's the right thing to say, the, the, in order to get to that point, I had to make some, some non-degeneracy assumptions. And it's conceivable that, and I've not completed this calculation, it's conceivable that, that there are some degeneracies that show up here that are not captured in the S1 equals 2. That's, that's work in progress. But there are a lot of these, there are a lot of these metrics that have their curvatures invariant under a circle, under that circle action. What's even more interesting is that the, there's one other particular circle that has a lot of invariant curvature tensors, and that's the thing of slope one. The circle of slope one P and Q are equal. And that turns out to have a seven dimensional family of invariant, of invariant curvature tensors. Again, it preserves the splitting C plus C plus C acts irreducibly there. And there the count appears to be S1 is four. Again, they're the same degeneracy cases that show up here show up in the seven case. It's not too surprising. But, and they still need to be resolved. But again, you get, there is a, there is a general family that depends, that's, that's very, fairly flexible in the, in the slope one case. We still don't know anything about completeness. These things, the, the solutions actually show up are, can be described in terms of, of families of pseudo holomorphic curves in an almost complex manifold that, that we, we don't have any good understanding of its completeness properties right now. Just to, just to do the, do the next interesting case, the, the case in fact that, that I was, that I wanted to understand because you may know we are, I and Simon and, and, and eight other of my PI's are involved in a Simon's collaboration on special holonomy in, in geometry, analysis, and physics. And, and part of our, part of our project is to understand special solutions of the, of the, of the G2 holonomy equations. And so, a natural thing to do, and this was what really got me started on the project, was, was looking for ways to find new solutions, new global solutions. And in fact, what Simon and I had done years ago, I guess it was in 1987, was we had looked for a co-homogen 81 solutions and, and you know, again, if you look at the list of subgroups of G2, there they are, closed subgroups of G2. There's, there's quite a long list, but of course the first one if you look at SU3, that's a maximal subgroup, if you look for, there, it doesn't leave invariant any curvature tensors, it doesn't leave any invariant any G2 curvature tensors at all. And so, and so, so the only case that could possibly occur there is the flat case. But for SO4 and U2, one, one of the, there are two non-congruent U2s, non-conjugate U2s in G2, that, but for these two particular ones, although we didn't think of finding them this way, this is what they actually turned out to be, it, we, we had constructed G2 holonomie metrics on the, on, on the self-dual bundles of S4 and CP2, self-dual two-form bundles of S4 and CP2, and it turned out that their curvatures took, take values exactly in these, in these special, the, these special orbits in the, in the KG2 that have non-trivial stabilizers. So, that, you know, from this point of view, those two things show up completely naturally without assuming, without assuming coal-mogeneity one or anything like that, just assuming a natural condition on the, on the curvature tensor. If you look at the other U2 that's, that's not conjugate to this U2 sitting in G2, it turns out that these guys don't exist, even though they have the same dimension space of invariant curvature tensors, it turns out that this, this U2, which acts preserving a line, a two-plane and an R4, that's incompatible, there are no solutions there. In T2, there's a, the maximal tourist in there, there's a five-dimensional space of invariant curvature tensors, and it leaves only constants. I, I have by no means completed the analysis and all these things, but some interesting things have happened. I mean, like cases like this where, where surprisingly you don't get any solutions, but again, when you get down to the torus, to the circle actions, it turns out that you get, you get solutions that, that appear on, that appear on arbitrary, depend on arbitrary functions of one variable. This one is a complete mystery to me though. The, the, if you look at this particular circle sitting inside the, all these circles of course, sitting inside the maximal tourist, which sits inside the issue three, so you can think of them as, as the, the PNQ have the same interpretation as before. And, and this particular guy, this slope zero one, it turns out have an enormous number of invariant curvature tensors, and, and understanding what its space of solutions looks like is at the moment, at the moment beyond my technology. But we do have, I mean, we do have kind of a, a plan for completing this. It's just at the moment there's still, there still remains several cases to resolve, but the cases that we have resolved or, or even partially resolved have already given us some interesting, interesting new, new geometry and new metrics. Let's see, yeah, I don't have that much longer to go. For part two, and I, so I may have to, I may have to sort of narrow what I'm saying, but I just wanted to point out that the analogous thing that, that you can look for, you know, Lawson and Harvey introduced the notion of calibrated geometry. I mean, wrote a fundamental paper on calibrated geometries in the early 80s that was largely inspired by, by understanding these geometries and special dimensions that, because they were, they were special because of the holonomy, this G2 and spin sudden holonomy. And they pointed out that they were connected to various minimal submanifolds, the, the so-called associative submanifolds being the, for example, in R7, which are exactly, you know, if you look at G2's action on, on R7, acting irreducibly on R7, if you look at the, at the planes, the three planes that are equivalent under G2 action to R3, that sweeps out this famous, this famous symmetric space of G2 sitting inside the gross monion of oriented three planes in R7, the associative gross monion. And, and as Harvey and Lawson proved, this, this any, any, any three manifold whose tangent plane is everywhere in an associative plane is in fact not only minimal, but absolutely minimizing in homology with respect to fixed boundary. And, and of course this generalizes to, to arbitrary G2 manifolds, the notion of an associated submanifold. In the general case, if you have a group acting by isometry as an RN and you have a G orbit of, of M planes, you can talk about sigma manifolds whose tangent places, spaces belong to sigma. And the, and of course the Gauss map then maps into sigma and it has a derivative which is, maps the tangent plane into the tangent plane of the gross monion of the, the sigma which is, and the tangent plane is, sits inside the normal bundle tensor, tensor, the tangent bundle, which looks like this. And because of symmetry, this, you can think of the Gauss, the derivative of the Gauss map as the second fundamental form. It actually sits inside this space, the tangent space of sigma tensor, the cotangent bundle intersected with the normal tensor SIM2. And so you should think of this intersection as the space of possible second fundamental forms for sigma manifolds. G is compact. In this case, I'm going to, I'm going to say G is acting by isometry as it's a closed subgroup and it's compact. You get a, for this class of submanifolds generalized in the associative case, you get a notion of, you get a notion of what are the possible second fundamental forms. And in general, these, all these spaces of course, the, the subspace RM is perpendicular and the tangent space of the gross monion are all H modules. So this guy is an actual, it's a representation space of H, the set of possible second fundamental forms. And so for example, in the case of special Lagrangians, where G is, G is SUM and H is SOM, then, so these are the, the orbit of RM under this action is the space of special Lagrangian tangent planes. And there it turns out you can calculate that the space of second fundamental forms is exactly, is exactly the traceless cubic forms in M variables. And, and for example, in M is 3, this is R7. And, and what you can do is look for, look for special Lagrangian threefolds, which is a, you know, this is a nonlinear PDE that's in fact quite difficult to understand what its singular solutions look like and what kinds of singularities they can have and so forth, which is something of intense interest in our, in our understanding of, of associative sub manifolds of, of G2 manifolds. And, and if you look at the, if you look at the space of, of Lagrangian threefolds whose second fundamental forms have nontrivial symmetry, that's the analog of what we were discussing for non, for curvature tensors with nontrivial symmetry. We find out that there are lots of many, that there are many integrable cases and connections with, connections with integrable systems. My student, Marianti Enel, did a similar case for the, for the M equals four case and found many integrable cases there. I want to just say what happens in the associative threefold case. If you look at, here G is G2, the stabilizer of a three, of this three plane is a copy of SO4, as I was saying. So the Grosmonian is, is, the associative Grosmonian is G2 mod SO4. The stabilizer, it's spin four, so it's SP1, SP1. So the, so it's not hard to calculate that the, that the tangent plane is in fact the, the three one irreducible representation for SO4 using those as the two maximal tori, which is SEM3 of one of them, tensor over the quaternion, they're both quaternionic vector spaces, tensor the fundamental representation of the second one as the eight dimensional space. When you calculate the second fundamental form as a representation space, it's, it's five dimensional, I mean it's S5, the tensor V01. And so it's a 12 dimensional space. And, and, but the point that you want to, you want to carry from this irreducible, from this realization of it as a representation space is that you can think of the second fundamental form as a quintic polynomial in two complex variables. It's a, I mean as an abstract space this is kind of hard to understand what the action looks like. But it turns out that if you interpret the second fundamental form as a quintic polynomial in two complex variables, then the second fundamental form up to, at a point up to a multiple can be thought of as a degree five rational mapping of Cp1 to Cp1 up to isometric actions on the two Cp1s. That is, you want to understand the degree five rational mappings up to isometric rotations in the domain and range two spheres. That's what the, that's what the, the SP1, SP1 act as isometric rotations in this, in the two Cp1s. And this geometric picture of it makes it easier to bring algebraic geometry into the picture. Let me, I'm going to skip the co-associative case. We didn't talk about that anyway. So anyway, we can work out the stabilizer types in the, in the associative case. If, if you have one of these associated second fundamental forms with non-trivial stabilizer, then the orbit of P is one of the following. It could be a quintic power. It could be, could be a fourth power times, times a linear term. Could be cubic and, cubic and quadratic. All of those have a circle as a stabilizer, a different circle, but a circle is a stabilizer. It could have stabilizer Z4. It looks like this. It could have stabilizer, stabilizer Z3 if it looks like this. And it could have, and I was noticing when I was looking over my slides earlier, that should be a two there, not a, not a three because it's got to be a quintic. It could have stabilizer Z2. Those are the possible stabilizer types. And, and there are some inequalities you have to be sure. For example, you don't want B to be zero because then that puts you back in case one, things that there's some inequalities to make sure the stabilizers stay exactly that. But I won't go through the details. And, and like I said, these three circles sitting in SO4 are all non-conjugate to each other. They, so they correspond to completely different geometries, even though they, the stabilizers all look the same. So for example, it turns out that the, the associated three-folds whose second-voluminal forms have this type turn out to be ruled surfaces. They depend on six functions of one variable. And they can be interpreted as pseudo-holomorphic curves in an almost complex three-manifold. In fact, they correspond one to one with pseudo-holomorphic curves in an almost complex three-manifold. And more generally, if you look at for ruled associative three-folds, it turns out that they can be interpreted as pseudo-holomorphic curves in naturally in a almost complex structure on the, on the space of lines in R7. I don't have time to talk about that, so maybe I'll skip that. It turns out that if you go to the second type, the second stabilizer type that has a circle symmetry, it turns out there are none of those. That's, again, it turns out there's no, the compatibility conditions, you know, when you differentiate, eliminate those cases. On the other hand, if you look at the third type that has a circle symmetry, they actually turn out to be special Lagrangian in some R6 sitting in R7. And in fact, they're the Harvey Lawson examples with an SO3 symmetry. So even if you hadn't known about those to begin with, they would, they would have been found by this process of looking for things with non-trivial stabilizers in their second fundamental forms. In Z4, it turns out that these actually have that, you know, that if the second fundamental form looks like this everywhere, then either A is zero or B is zero. And so, and so they, and so you don't get anything beyond the two cases that we already discussed. And even the A equals zero case puts you back in Proposition 2 that says they don't really exist. The Z3 case, that case, it turns out that those actually turn out to be special Lagrangian also in R6. And they were classified in this 2000 paper, the associated three-folds that have a Z3 symmetry because it turns out that as special Lagrangian things, they have a Z3 symmetry. And they were classified. Z2 is the hardest case. There's still, I'm still trying to finish the calculations in that. There are plenty of examples. There are special cases, but it breaks up into a large constellation of degenerate cases that I have not yet tracked down. So, I can't tell you what happens there. But again, it breaks up into a collection of different integrable systems that we had never seen connected with, connected with associative sub-manifolds before, associative three-folds in R7 before. And exploring that connection is one of the things that I and a couple of my graduate students are working on now. Yeah. So, the, I would say the take-home message is that there's, that if you look for geometric conditions, that is, algebraic conditions on the curvature tensors that are inviolative and compatible, it's a natural way of finding connections with integrable systems. And the, right now, I think we only have scratched the surface in understanding that connection. And being able to find out which of those things are compatible with complete solutions is another story that we, that we still don't understand completely either. But it's, to my mind, a promising way to understand these reductions to integrable systems that turn up every now and then. And it also strikes me as sort of in the spirit of Berger's classification of hole only. In that, in that there's a clear, there's a clear set of problems that you need to go through the list to get things. And when you, but when you come to the end, there's an interesting list of solutions. And they were completely unpredictable until you actually sit down and do the calculation. That's so far, so far we're, you know, it's something we're exploring. And, and I, my fondest hope is that one, one day, it will turn out to be as beautiful a story as the hole only story has been. And, and I think my time is essentially up. I'll stop here. Thank you. I have one. Do you recognize an integrable system there in this setting? Yeah. So when you say recognize an integrable system, the, there's, you know, what's the best thing to say? You know, when you talk to integrable systems people, they always, the fundamental question is what is an integrable system, right? Yeah, there's a, there's a book written by the experts called what is an integrable, what is integrability, right? This is, you know, in some sense, integrability is, is, you know, one thing is that you can, you can actually sometimes make a direct comparison between the integrable system, the system you've found and a known integrable system, for example, sign Gordon or cinch Gordon and so forth. That's one thing. And the other, the other interesting case is that, is that you find, see all of these cases that turned out that with S1 is some positive number and all the higher ones are zero, those all reduce to, those all reduce to systems of PDE on a Riemann surface, ultimately. And these systems of PDE are over determined and you wouldn't expect them to have solutions, but they do. And it's, you know, while I can't in every case write down a connection with a spectral curve, which is one of the things that the integrable systems people really like is their, their method of, of, their method of bringing it into the integrable fold is to, is to, is to write down a nonlinear eigenvalue problem and, and get a spectral curve and that, you know, that gets attached to the Riemann surface in a, in a specific way. And, and whenever that happens, they say it's integrable. But in this particular case, it's not, it's not, it's not the, the strongest form of integrability where you can actually say I can write down all the solutions in terms of holomorphic functions. That is, that's the, that's the, that's the Louisville phenomena. But it's sort of the next stage that you, that you have an over determined system where, where the, that you wouldn't normally expect to have solutions. And, and yet they're compatible. You were explaining that when you have this R7 and you were reducing with respect to group SO4 and U2SO1, you have this self dual bundle over this S4 and CP2. These are the connections of whatever you were saying with integrability, these ASDs and young mill self dual equations as well as your holonomic conditions there. Actually the, there are, there are ways to write down solutions of the, of the holonomy equations in terms of solutions of Yang-Mills equations. The, there, there's a, right. But, but this particular, this particular writing down the issue, writing down the, the G2 metrics on lambda 2 plus of S4 and CP2, that's of, you know, that wasn't done by dimension reduction in that sense to a, to a Yang-Mills system. It was actually reduced to, reduced to an ODE system. You have this, there's a paper where Athea and Pot, they actually, yeah. Oh, yeah. That, where they, where they write down the, where they write down solutions to the, to the self dual Yang-Mills equations in terms of, in terms of a classical integrable system. That's a, you know, it's a, that's a different phenomena from what I'm describing here. But, or, or as far as I know it's different. There, there may be a connection that I'm not aware of. You said the word about the getting complete solutions. Do you, do you have any idea whether there will be one of the cases which will be definitely simpler or for which you have maybe other tools to get to? Let's see. Well, to, to prove completeness in the cases where I, where I can prove completeness. It's because, it's because when you write down those structure equations as I was showing in the, in the, right at the beginning at the, for the, for the SU2s with a double eigenvalue, what you need to do to prove completeness is you need to prove that, that certain covariant derivatives are bounded by other covariant derivatives to guarantee that nothing blows up in finite distance. And for that, you need to, you need to actually get a formula for those covariant derivatives. Because those formulas will tell you what the, you know, once you have all eight quantities, you know what all their derivatives are. So if you want to show that something's not growing too fast, you basically what you want to do is prove, you have this, this, basically you have a vector field that you're trying to, trying to show it's complete in this r8, in this, in this eight dimensional space of invariance. And, and for that you need, you know, something like a Lyapunov function to guarantee that you, guarantee that you can't run off to infinity in finite time. And, and in that case you have, we have the, I didn't put up the formulas because they're, you know, at first glance, you can't tell what they mean. But in that case, you can actually use those formulas to get such a function to tell you when, tell you when the, when the curvature and its covariant derivatives stay bounded in finite distance. And that coupled with Carton's existence theorem is enough to show that those things are complete. They might be complete Orbefolds that, that you, that you can't rule out. They, they might actually, they might develop Orbefolds, Orbefold singularities, but, but they won't develop, you know, they'll, they'll be locally smooth and complete. Yes, all of these are non-compact. Any other questions? Back to the SU2 case, the drawing is still on the board. You made the claim that you cannot prescribe a, a laboratory curve as a value. Yeah. But there's still, maybe you cannot, but, but there is still maybe a family which you can deal with, which is like something close to the cusp since you have this result. And then maybe you can hope for some continuity result in the Carton system. Yeah, that's a question. Yeah. I mean, the, the thing is that the curve, any curve that admits a non-trivial solution has to satisfy a set of differential equations in this space, right? And, and we don't know, we don't know, yeah, we don't know enough about what those solutions look like to be able to, to be able to say that we could make something close, something that looks close to the cusp. Oh, oh, if it was, if it was non-compacting, well, non-compact always occurs, always, no, that's not true. The algebra is just using some continuity in the system, the Carton system. Yeah. Well, you do have continuity in the Carton system. If you get one compact solution, then you can, then you can get, for a given algebraic curve, then you can get that the nearby ones are compact, right? I mean, not compact, complete. Yeah, that's right. If you, if you get, if you get one of the, that's right. If you get one complete, then you can show that the nearby ones are complete. That's right. Yeah. Okay. If there are no more questions, thanks for a great talk. Thank you.