 I'm happy to listen to you. Great. Thank you so much for organizing such a wonderful conference. Appreciate it. The title of my talk today is the same as the title of a paper I've been working on with Thomas Barrett, but really I'm going to talk about a larger body of work that I've done with Hans Halverson and Jim Weatherall along with Thomas and some of the papers leading up to this one and some papers that have come after this one and some that we're working on now. Okay, so in the second chapter of his book, World Enough in Spacetime, John Ehrman constructs an elegant hierarchy of classical spacetime theories. And the hierarchy tracks both the geometric structures involved, but also the associated spacetime symmetries. And he finds that as the spacetime structure becomes richer, the symmetries become narrower, and the list of absolute quantities increases, and more and more questions about motion become meaningful. Following Ehrman, I will present a hierarchy of spacetime symmetries today, but instead of comparing the symmetry properties of different, excuse me, of different spacetime theories, I'm going to restrict attention to general relativity and compare the symmetry properties of different spacetime models within that theory. So the hierarchy turns out to be analogous to, say, the hierarchy of causal conditions that's long been used in the foundations of general relativity. And I'm going to highlight a few connections between this symmetry hierarchy and a hierarchy of causal structure. Let me just say that there's many levels to the symmetry hierarchy. I'm only going to, because of time, I'll only be able to focus on four today. So here they are. These are the four that I'll be focusing on. And I'd like to emphasize that as we move up the hierarchy, the symmetries decrease, and so the structure increases. And it turns out that with this additional structure at each level, one is able to prove various types of things. And so again, this is kind of analogous to the hierarchy of causal conditions. So I'll take one example. Somewhere up at a certain level on the hierarchy of causal conditions, one is able to prove that the causal structure of spacetime is rich enough to, say, determine the topology of spacetime. This is a classic result by David Malament. It's actually going to be coming up a bit later in the talk, but that gives you an example. So as you move up the hierarchy, one is able to do more and more. And in just the same way that with globally hyperbolic spacetimes, one is able to prove an awful lot more than if one is just considering chronological spacetimes. Each of these four levels corresponds to a recent paper that I've written with either Hans or Thomas or Jim, and here they are. I'll also be talking about some future work with Thomas and Hans. Okay, so let's start. Let's consider some preliminaries concerning isometries. I just want to make sure that we all have a good idea of the definitions. There's only a few definitions, but I think being very, very clear about what one is saying is important. So here's my definition of a spacetime. It's a pair mg, where m is a manifold, g is a Lorentzian metric. The usual background assumptions concerning the manifold are in place. It's smooth, it's Hausdorff, and so on. We say that spacetimes mg and m prime g prime are isometric if there's a diffeomorphism from m to m prime such that when you push forward g, you get g prime. And let's say that the collection of isometries from a spacetime to itself are the global cemeteries of spacetime. So again, one could possibly explore many different types of definitions of a global cemetery of spacetime. I want to be very clear, this is what I mean by a global cemetery of spacetime. And it's the one that Wald uses in his book. It's very influential within that literature, Garrosh, Penrose. It's also the one that Erman uses when he's constructing the hierarchy of cemeteries in classical spacetime structure. So again, been very influential within the philosophical literature. So I understand there might be other definitions of cemeteries out there. I'm focusing today on this one. Now let's consider a very influential construction used in discussions of the whole argument. So let mg be a spacetime and let h and its complement have non empty interiors. Well, we know there is a whole diffeomorphism from m to m, which is non trivial in the whole, but acts as the identity outside of the whole. That's what I mean by a whole diffeomorphism. So a trivial proposition one can prove is that let mg be any spacetime and let psi be any whole diffeomorphism. Then the spacetime mg and the spacetime m where you push forward g, those models are isometric and the isometry is realized by psi, the whole diffeomorphism. So here's the picture. I think we're all familiar. Okay, now for the quiz. Erman in his second chapter has a quiz at the end. Here's my quiz. And yeah, I'll just say that in the past few years, I've had many email discussions, many discussions via referees, many discussions in person, where I come across people holding views consistent with A, B and C. I'd like you to think about the question for a minute here. A whole diffeomorphism is blank, an isometry from a spacetime to itself. Is it always an isometry from a spacetime to itself? Is it sometimes an isometry from a spacetime to itself? Or is it never an isometry from a spacetime to itself? Okay, the answer is C, never. So a whole diffeomorphism is never an isometry from a spacetime to itself. Trivial doesn't count, I take it. Trivial doesn't count, yeah. But a trivial would be not a whole diffeomorphism, because by definition, I have to move something non-trivially in a whole diffeomorphism. This follows from a very general uniqueness result that's been around since 1969 from Garrosh. And a simple corollary is that a whole diffeomorphism is never a global symmetry of a spacetime. Okay, let's now consider the hierarchy. Let's start with the bottom and move our way up. So the weakest condition in the symmetry hierarchy draws on the whole construction just considered. So the condition essentially requires that when the global symmetry of spacetime are fixed in an open region, however small, then they're fixed everywhere. So following Garrosh, we'll refer to these spacetimes as rigid to avoid confusion with the whole argument. So we say a spacetime is rigid. If for any open set in the spacetime and any isometry from the manifold to itself, then if psi is the identity on the open set, then it's the identity everywhere. And so we show that every spacetime is rigid. Now, despite the fact that every spacetime is rigid, you might get the sense that this is an extremely weak condition. I think it's fruitful to explore situations where spacetime is not rigid. And so here's another proposition that's related. Non-Hosdorff spacetimes always fail to be rigid. So here's the picture. Consider Minkowski spacetime. And you've got two origins that are non-Hosdorff-related. These are the witness points witnessing the non-Hosdorff-ness. And now just consider a bijection, which takes each point into itself, but exchanges the witness points. That bijection turns out to be not only a diffeomorphism, but an isometry. And so fixing the global symmetries outside of a hole, which contains the witness points, will not fix what happens inside the hole. So the spacetime is not rigid. Or you could say that non-Hosdorff spacetimes have a certain type of symmetry holes that standard models just don't have. Okay, now let's talk about the next level. So the next condition in the symmetry hierarchy requires that the global symmetries of spacetime just be completely fixed. One doesn't have to fix an open set somewhere to fix it everywhere. It just comes already fixed. So we say spacetime mg is giraffe if the identity map is the only isometry. So here's a way to think about this. David Malament suggested one way to construct a giraffe spacetime. You take Minkowski spacetime, you take a compact region shaped like a giraffe, and you delete it from the manifold. So here's a giraffe spacetime. Here's another giraffe spacetime. This one can work with it. It's easier. It's a portion of two-dimensional Minkowski spacetime. You just cut this little triangle out, delete everything else. Now you've gotten rid of all global symmetries. The only isometry from this spacetime to itself is the identity map. Of course, any giraffe spacetime is rigid. Every spacetime is rigid. The other direction does not hold. Minkowski spacetime is a counter example. All sorts of symmetries, global symmetries. So it's not giraffe, but of course it is rigid as we've seen. I want to emphasize that virtually all example spacetimes you've ever seen in your life are not giraffe. And yet everyone knows that giraffe spacetimes are actually generic among spacetime. I think that tension is something interesting to just meditate on because I think we're accustomed to thinking about spacetimes with symmetries when our universe probably has none. But the meaning of generic in this context is never made precise and in general proof does remain elusive. Partial results are available namely on compact spacetimes due to mountain. All right. Time for the next level. The next condition in the symmetry hierarchy requires that the local symmetries of spacetime be completely fixed. And I'll say what I mean by that here. So we say that a spacetime is Heraclitus. If for any distinct points P and Q and any open neighborhoods around these points, there is no isometry from one neighborhood to the other, which takes P into Q. So in other words, we see that since any neighborhoods of distinct points fail to be isometric, we have a sense in which each event is completely different from every other event. And so you might say that in such a world it's impossible to step into the same river twice. Okay, one can show that any Heraclitus spacetime is giraffe, of course. The other direction does not hold. So back to this example, we see that of course it's giraffe, it is not Heraclitus. This is a flat spacetime. So every point is exactly like any other point in terms of there is a local isometry there that one can set up. So not only does this spacetime fail to be Heraclitus, it sort of maximally fails to be Heraclitus. One can show that Heraclitus spacetime does exist. So let me say a bit about this. It's good to have a feel for what a Heraclitus spacetime looks like and feels like what a concrete example certainly helps me. So start with a portion of Minkowski spacetime and standard TX coordinates and call it MG. So this is the portion I have in mind. It's back to that triangle. Down here at this point, this is the origin in a standard coordinate system from Minkowski spacetime. So I'll be referring to the origin, but the origin of course is not included in the spacetime. We've deleted it. Now what you do is you consider a scalar function, which is the Euclidean distance from the origin. And then a conformal factor where the conformal factor is just one over that Euclidean distance from the origin. Now consider this conformally flat spacetime. The claim is that it is Heraclitus. So how do we know that? Well, we use scalar curvature invariance to do this. So one can show that the Ricci scalar has this expression here. And it follows from that that any local isometry must map P to a point Q with the same Minkowski and distance from the origin. And so the picture is that if I'm considering a point Q and I want to map it, even locally map it to another point in the spacetime, I'm constrained to map it where my Ricci scalar is the same. And the Ricci scalar is the same only on that blue line. Now I construct a different scalar. I'll call it Q, where I just take the covariant derivatives of R and use the inverse metric to form a scalar function. And one can show that Q takes this form. It's proportional to both R and F squared. And so you can work out the details. It follows from this that any local isometry must map P to a point Q with the same Euclidean distance from the origin. So here's the picture. The point must be mapped somewhere on this red line if the F values are going to be preserved. And so taking these two constraints together, then if you're considering a point, you are forced to, if there's a local isometry, you have to map the point to itself. And that's true for every point in the spacetime. And so there you get that the spacetime is Heraclitus. So you're actually proving something much stronger than the Heraclitus property there. I'll say a bit more about that in a moment. But I have a question, which is, do four-dimensional Heraclitus spacetimes exist? We've constructed a two-dimensional one. Man, I've tried. I really have tried. I can't figure it out. There doesn't seem to be any reason why you would have a block here to be able to do it. It's just things get really complicated. Presumably one would have to find four different scalar curvature invariants and play them off of each other like coordinate systems like I've done in the earlier example. So really what's going on there is that my curvature invariants are, yes, playing the role of coordinates. And so I use those coordinates to prove this Heraclitus property. And yeah, so can one do this in higher dimensions? But Heraclitus spacetimes are highly structured at each point. So this allows for some pretty crazy uniqueness results. Remember, as we're going up the hierarchy, the structure is increasing. This allows us to prove more and more results. So here's something that you can show at this level of the hierarchy. Let's start with a couple definitions. A pair of spacetimes are locally isometric if, for each point in the first one, there's a neighborhood around the point that you can embed into the second one and vice versa. And then we have to define what a local property is. Let's say a property of spacetime is local. If given any two locally isometric spacetimes, one has the property if and only if the other does. So then you can show that Heraclitus spacetimes are locally isometric if and only if they are actually isometric. That's kind of crazy. So a corollary there is given any collection of local properties, there is at most one Heraclitus spacetime with exactly those properties. There's a uniqueness result here. It's telling me you don't have to tell me what the global structure of your spacetime is like. You just tell me the local structure at each point and I'll be able to put the puzzle together in a unique way. If it's possible, of course, it may not be possible. So one could rephrase this by saying for Heraclitus spacetimes, local structure just determines global structure. And of course, we have the other way around global structure determines local structure. And therefore, these two structures are kind of similar in this way when one can recover one from the other. Here's another uniqueness result. So consider a spacetime MG with any property P. We say that this spacetime is P maximal or P in extendable if the spacetime cannot be properly and isometrically embedded into some other spacetime with the same property. And it would seem that Leibnizian metaphysics would demand that spacetime be maximal with respect to various physically reasonable properties. And now consider this definition of observational indistinguishability. One spacetime is observationally indistinguishable from another. If for each point in the first, there's a point in the second such that their causal paths are the same. There's an asymmetry in the definition here, which will be important in a moment. But there is a general underdetermination problem when one considers the class of all possible spacetimes. But you can show that if one restricts attention to Heraclitus maximal spacetimes, so these are spacetimes with the Heraclitus property, which are also maximal with respect to this property, then these spacetimes are observational indistinguishable if and only if they are isometric. So we have Heraclitus asymmetry, Leibnizian maximality gives us no underdetermination problem. Or another way to put the point, for Heraclitus maximal spacetimes, the observable structure determines the global structure. Okay, and now finally to our highest level of the hierarchy. This condition requires spacetime to have maximal curvature structure. So for any manifold, you can consider the collection S of M of smooth real valued scalar functions on M. And associated with each spacetime, MG is a collection CMG, which is a subset of S of M. These are the smooth invariant scalar curvature functions constructed from the metric and its associated Riemann curvature tensor and covariant derivatives. And now we can consider the following definition. Let's say that a spacetime is maximally structured if the collection of invariant scalar curvature functions just is the collection of smooth functions. In other words, every smooth function you can possibly imagine on the manifold corresponds to some invariant scalar curvature function. So one can show that any maximally structured spacetime is Heraclitus. Question, does it go in the other direction? I don't know. It would be cool if these two conditions were turned out to be equivalent. But yeah, this question is open. Now maximally structured spacetimes do exist. And so here's a sketch of the proof. Consider the Heraclitus spacetime that we already took a look at in the previous section in standard TX coordinates. This is just, again, conformal Minkowski spacetime. Use invariant scalar curvature functions R and Q that we already took a look at to define invariant scalar functions T and X where T and X are, they mimic the coordinate maps, the global coordinate maps. So then you consider an arbitrary function on M. You put that function in coordinates T and X, you can reconstruct that same function by using the invariant scalar curvature functions big T and big X in place of the coordinates. And so you can build your smooth invariant scalar curvature function, which just is whatever F you gave me to start. So at this highest level of the hierarchy, some even stronger uniqueness results are available. We know there are a number of senses in which the scalar curvature invariance can be used to determine the local geometry of spacetime. There is a literature on that. But in general, one finds that such functions do not determine the topology of spacetime. For example, one finds that local invariance cannot distinguish a plane from a flat torus. So let us make this claim precise. Let Mg and M prime G prime be spacetimes and let Mg and C of M prime G prime be their collections of invariant scalar curvature functions and let them be indexed by a set I to generate the respective families of functions Fi and Prime I. And we do this in such a way that for each I, Fi and Fi prime, they correspond to the same metric construction. So however you're constructing your scalar curvature function from G, do the same thing with G prime. Now we say a bijection from one manifold to the other is a curvature isomorphism if for all, for any I that you choose, when you pull back the F prime, you just get F. So F prime of I pulled back just is F of I. And so you can show that of curvature isomorphism, it need not be a homeomorphism, let alone a diffeomorphism. So in this way, the situation is completely analogous to the one concerning the malament result. You set up a I'm sorry, a causal isomorphism and you show that at low levels in the causal hierarchy, a causal isomorphism just isn't a homeomorphism, but that at some level it is. So to show in this case that the result doesn't go through, just consider any bijection at all between any pair of flat spacetimes with different topologies. So for example, consider Minkowski spacetime and Minkowski spacetime with a point removed or a torus of some kind. One finds that when one pulls back any scalar curvature function from the one spacetime to the other, you get that the functions are the same. So you have a curvature isomorphism, but you do not of course have a homeomorphism since by construction, these spacetimes have different topologies. Well, you can show that if you're willing to go all the way up to maximally structured spacetimes, a curvature isomorphism not only is a homeomorphism, it has to be a diffeomorphism. So the result is completely analogous to the one from Malamint showing that sufficiently rich causal structure is going to determine the topology of spacetime. But I want to emphasize a very important difference between the cases. The causal structure of spacetime is encoded in a particular relation between spacetime points. Relations come to the fore. On the other hand, the curvature structure of spacetime is not encoded in a relation between spacetime points. Instead, each scalar curvature function gives rise to an invariant property of each spacetime point. And the scalar curvature structure of spacetime is encoded in the collection of all of these properties of all of the spacetime points. And so what I'm saying is that maximally structured spacetimes have so much structure that even the properties of spacetime points determine the topology of spacetime. You don't need any relations to do this. What justifies the label maximally structured? Here's some current work with Hans and Thomas. So given a maximally structured spacetime MG and any tensor field whatsoever, T on M, one can explicitly define T from G. So in particular, any global coordinate map that you might be interested in is explicitly definable from the metric. In addition, any other metric G prime is explicitly definable from your original metric G. And so we can say that for maximally structured spacetimes, all structures curvature structure and any structure can be explicitly defined from the local metric structure. Okay, let me just close by with a brief review of where we've come. We've looked at four conditions of a symmetry hierarchy or asymmetry hierarchy. As you move up the hierarchy, symmetries are reduced and structure increases. There are a number of levels in between, but there's nothing lower than rigid and there's nothing higher than maximal structure, at least that I have considered. Maybe there is. Of course, there is. Whether it's of interest is another matter, but at the lowest level, rigidity, this is the condition that holds when fixing the global symmetries of a spacetime in a little open set fixes them everywhere. And what we saw was that every standard spacetime in general relativity is rigid, but that you can find non-rigid examples of spacetimes if one is considering non-Hosdorf models. Then we bump up a level to giraffe spacetimes. These are the spacetimes where the global symmetries come already prepackaged fixed. There's only one isometry of a spacetime to itself and that's the identity map. And these spacetimes, of course, are rigid. The implication does not go in the other direction. These spacetimes are or it is thought to be generic in some sense. And yet most of the standard examples that one sees in the literature, whether that's textbooks or research literature, these spacetimes are almost always giraffe spacetimes. Then the next level we bump up to Heraclitus. Here's where we not only fix the global symmetries of a spacetime or the global symmetries come fixed, also the local symmetries come fixed. Heraclitus spacetime is one in which one can't map distinct points into each other even locally. And at this level, now there's tons of structure both globally and locally. And with this, especially the local structure, one is able to show that certain claims follow that didn't claim lower down on the hierarchy, but now follow. And one that we highlighted was you give me the local properties of spacetime and I'll give you the global properties of spacetime. We also highlighted another uniqueness result which essentially says that the observable structure determines the global structure. Finally, we jump up to maximally structured spacetimes. These imply the Heraclitus spacetimes, the other direction is unknown. So these top two levels perhaps may be equivalent. But at this highest level, we've shown that there's enough structure that you give me a pile of spacetime points and you give me their internal properties. You don't tell me anything about how they're related at all. I can use that internal structure to give you back the global topological and even the manifold structure of spacetime. And moreover, one can use maximally structured spacetimes to explicitly define any structure that you're interested in as long as that structure comes in the form of a tensor field. Okay, that's it. Thank you very much. Thank you. Yeah, it's a technical material, but you explained it very clearly. And now we have Brian Roberts, which is your respondent and he has up to 10 minutes to tell us something interesting about this. Well, I think JB has said all the key interesting things. I'll try to be really brief just to allow maximum discussion here. Thank you, JB. Wonderful. Wonderful. Lovely to see these things all tied together. I've been following these papers, of course, with interest. And so the hierarchy is just, I just want more, really. So my first comment is, there's another interesting work that wasn't mentioned here by JB and Jim Weatherall from 2014 regarding conventional, conventionalism about spacetimes. And so my first sort of question is, I wonder if you've thought about whether a relation of conventionality might fit into the symmetry hierarchy in some interesting way. Their thought, for those who haven't read this paper, first, you should definitely read this paper as well, amongst all the others, wonderful piece. Their thought is when one says things like I can buck you to like these two spacetimes, you know, you can't really tell which one you're in, because there might be funny forces which make you think there's a certain metric which there isn't. And then JB and Jim very reasonably respond, what do you mean by forces, my friends? And you ought to have some minimal reasonableness constraining what you mean by forces. And that would mean something like F equals MA or in a relativistic sense, you'd want a two form characterizing the force tensor field so that you have F equals MA in a relativistic sense along every time like frame. So and then the result shows that for the class of conformally equivalent spacetimes, there basically is no conventionalism. But of course, this condition can be relaxed, you can look, you know, for conventionalism and more general spacetimes. And anyway, I just sort of wonder how this could potentially some note, they also point out that depending on if you know how rigid you are about what you mean by force, there could be lots of other you know, interesting notions of conventionality you could identify, you just have to tell me why you think that notion of a force is interesting if it's not a rank zero to a symmetric tensor. So anyway, just looks to me like some room to play here. I don't, to be honest, see how they would be related by implication to many of the things you said, but I wonder if you thought about this. Some other very brief comments. The original Geroch stuff was about limits of spacetime. So I mean, I think one of the things that's being revealed here is interesting tension between these, maybe two research programs, research programs involving non house dwarf spacetimes, which you sometimes use for reasons of branching among other things. I mean, people sometimes appeal to this when they want two points to get really close, just closer than you can get in an ordinary spacetime. So branching could be a reason. But then, you know, these limits of spacetimes, of course, we have different options, but that rigidity result was used by Geroch is a key step in showing the existence of so-called Geroch limit spacetimes. And these are crucial in certain sorts of stability results, which, you know, help motivate what it means to be a generic spacetime. So I wonder, does attention between these two sorts of research programs, does that adopt the Geroch approach to limits? And people who might try to use non house dwarf spacetimes for interesting modeling and physics. Gosh, there's lots of other things to say, but I don't want to just carry on. Maybe I'll just leave it at that. And I mean, I'm sure other people have things to say about the whole argument. JB already knows what I think about that. So I'll leave it there. And thank you again, JB. Wonderful, wonderful talk. Yeah. Thanks, Brian, for your comments. I really appreciate them. I'll sort of try to give a response. Concerning the conventionality stuff. Yeah, you know, that I have never given that thought. That's intriguing. I do know that, you know, we're just piecing this structure together. It's a very big structure. You know, like I said, I think it's analogous to the hierarchy of causal conditions. One, one thinks about just all of the results that one finds there. I'm not on that level. I can't prove, you know, I'm not on the level of I'm in guzzy, for example. And so I'm very, very slowly working my way through this stuff and hitting mainly the really simple stuff. That sounds a bit complicated, but there might be something there. And with as the structure kind of builds out a bit, more and more results presumably will be collected. But I appreciate the heads up because I never would have even thought to put those two things together. About the limits of space time paper by Garrosh. Yeah, that's the context where this general uniqueness result shows up. And as you emphasize, Brian, you're absolutely right. This is the result that sort of forces the uniqueness of limits. One could take limits in all sorts of different ways. And it just turns out that the rigidity of space time in the sense that I've been talking about, which relies on the Garrosh stuff, picks out a unique kind of limit to look at. But you mentioned some tension. And so that's the part I'm still, I don't know what exactly tension, I want to get a better grip on that part. As I see my project, of course, it's no surprise. I just have these definitions around. And I prove things with these definitions. And so, of course, the mathematical theorems are true. And so there can't be any tension concerning the results, of course. So I guess I'd like a better picture of what that tension looks like. I sort of had a mind, sorry, if I wasn't clear, maybe the idea that if you take yourself to be modeling physical phenomena with non-Hausdorff spaces, and you want to say things like my result is robust under perturbations in the sense that there are nearby space times that all share the same property. That in non-Hausdorff spaces, I take it the lack of a rigidity theorem means there's a lack of a well-defined limit here. So it's not clear how long we get. Yes, absolutely. So I guess what I would want to say is we already had reason to be suspect of the non-Hausdorff thing. I have in my past work, I guess I've been playing that kind of open. I haven't come down one way or the other where I think, for example, a non-Hausdorff space time is physically reasonable or not. I've sort of left that door open. And to be honest, I've kind of emphasized the other side, which is probably what you're referring to. Now I can kind of see the tension. So I do have certain writings where I say, well, if you look at something like an initial value formulation, and it's in Misner space time, you can take a surface, put some data on it, evolve it to where you can't evolve it anymore. You have a maximal Cauchy development, and you see that in the Misner context, that maximal Cauchy development is extendable. And so then you're thinking, well, we can consider various extensions. And it just turns out that in that context, the maximal, maximal extension, when you're really thinking about it, what Garrosh calls the natural extension is non-Hausdorff because it sort of branches. And so exactly like you said, Brian, it branches in some sense. It turns out that that particular model doesn't branch in another sense. But it doesn't matter for this context because it is non-Hausdorff. And all you need is the non-Hausdorff property to show that the rigidity result fails. And so taking a cue from what you said, once you have that rigidity result failing, then now you don't have a good sense of limit. You don't have a good sense of generosity maybe. And so then how are you going to be talking about nearby possible worlds when you're considering a non-Hausdorff model? So I think you're absolutely right that these mathematical results are now coming down on the side where it's like maybe non-Hausdorff space times aren't as reasonable as we might have thought, right? And this is a reason why. Or to put it in a slightly different way, fixing global symmetries outside of a hole which contains these witness points, it's not enough to fix what goes on inside. And so there's a type of indeterminism that just is not in play in the standard models. And so if people are worried about certain types of indeterminism, then that's a type where you've got something going and you might become worried about that. And so I think you're absolutely right. This counts against non-Hausdorff models for sure. So thank you very much. Okay. Just thanks for the talk and for the response. And we can still take some questions. I wanted to ask about this issue of generosity. And as you correctly say, too much any space time you write down, he's going to have some symmetries, but we can regard that as non-generic. I mean, here's the sort of physics intuition reason to think that that I've done with would be something like, well, you know, take the solution, linearize around it, look at small, you know, for the gravity, for a little pulse of gravity radiation pretty much anywhere, and it's going to break the symmetries. And so the, you know, I can think in a kind of intuitive sense of the topology. I get an arbitrary small distance from any symmetric thing, and I'll find a non-symmetric thing. I expect the symmetric points to somehow be topologically sparse. What are the problems in rigorizing that way of thinking? Is it just the infinite dimensionality of it that becomes troublesome? Yeah, what comes into play I think is what troubles all sorts of foundational results concerning generosity and general relativity. One can put, well, first of all, one would like to be able to take the entire collection of all space times and put a topology on that collection. That turns out to be really, really problematic. And so instead what practitioners do is they'll focus on collections of space times having the same manifold, and then they'll put a topology on that. But the problem is, is that there are a number of different topologies one might consider, and none of them seem completely satisfactory. So that's a point that Garrosh made very early on when the topic of generosity was brought up, when one had to prove things that were generically true. It's also something that Sam Fletcher, I mean, that was kind of his dissertation, is to pull out reasons why there just isn't a, you know, a canonical topology to work with. And it just turns out that if one works with the compact open topologies, then results, of course, are easier to find in terms of stability results. But the physicality there comes into question. When one moves to the open topologies, the Whitney topologies they're sometimes called, then it's much harder to get results. And I think that's the problem here with trying to prove that giraffe space times are generic. The compact case is easy because in that case, then the two different topologies actually coincide. So the physicality of the compact topologies there. And so one can prove this result. My sense is, and this is not really something I've looked into much, I know Sam was interested in this question of proving generally that giraffe space times are generic with some interesting topology. I don't know of any work that's done other than this result that I cited from Mounend in 2015. Thank you. Okay. Any other questions? Yes. I don't see. Thanks. That was really nice. I actually just wanted to hear, I find this hierarchy of space time structure intrinsically interesting. But I'm wondering if you could say, do you think there are sort of philosophical upshots to be had? Do you adhere to various methodological principles involving relative amounts of structure? Or do you think there are some theoretical virtues that come from considerations of comparisons or structure? Or I'm just wondering where you go from here within philosophy of physics? Yeah, yeah, you're trying to get me to say stuff that I don't want to say. I want you to say it. That's the thing is that my project is one in which, yeah, I'm considering these very clear definitions and seeing what followed. There's already so much to do there. Of course, there's all sorts of interesting topics that are related to this stuff. And I'm certainly interested in that. But I just keep my cards pretty close to my chest. And I just see no need to wait into those waters when I've already got so much to do here. Sorry to sidestep your question. Okay, other questions? I would like just to continue this. Maybe you don't like, but you are speaking about space but what about matter and what about equations? What do you mean what about matter and what about equations? Yes, so for example, matter should determine the metric. So is this somehow represented in your case? Or for example, some Oxford persons, they think that the symmetry of matter determines the symmetry of spacetime. But you are speaking about the symmetry of spacetime as if matter was not there. So what do you think of these kinds of dependencies which seem absent in your talk? Yeah, let me say something about that. But before I do, I want to push back on something that you said. You said that matter determines the metric. That's just not right. The metric determines the matter, but not the other way around. There's an asymmetry there. Yeah, but if you have a massive thing, then it will have a curvature. If you have no matter, then you can have a flat spacetime. Yes, but you can have two different vacuum solutions on Rn, different metrics. So without matter, you can have differentiation. But if you have matters, then you have a curvature. No, I still want to push back on that. I'm sorry. Where is that theorem found? Maybe I'm not as specialist, but you're speaking about generality. It's not true that as soon as you have a massive matter. Yeah, so I guess what I want to say about matter is in the background, of course, one has Einstein's equation. If you give it a metric, then I can trivially define via Einstein's equation a matter field, a TAB field, the stress energy field. And so every spacetime is a solution of Einstein's equation in that sense. And so if one is considering a more restrictive sense, then you have to say what that means. So sometimes folks talk about exact solutions, or sometimes one might use energy conditions and so on to kind of limit what you're calling a solution there. But my results here hold generally. And then if you want to impose any constraints like that on top, everything will go through. So nothing I've said here will be contradicted by that discussion. It's a sub discussion of this one. Yes, okay. But you can have different distributions of matter, but you mean they will be the same metric? Will they be compatible with different spacetimes? I guess I'm not sure what you're asking. I mean, Einstein's equation is one in which if you give me a metric, I can come up with a curvature tensor. And I can use that to define one side of Einstein's equation that contains the curvature structure, a certain type of curvature structure. The other side is the matter structure. And so I can look for solutions in a variety of senses. Like I just said, every metric is going to produce some matter distribution via Einstein's equation. Whether one wants to count that distribution as reasonable or not is a question that you can check out. But at this level, I like to think of Einstein's solutions as coming in levels. I'm working on the most general level so that my results are the most general. If one wants to impose any kinds of constraints on matter, like it seems like you might want to do, then I invite you to impose those constraints. And then you'll have a limited discussion. But everything I've said will carry through. Yeah. Just in some discussions, matter is supposed to be primary. And what you tell would be coming afterwards. You have a reverse direction. Yeah. But I don't want to let this go, because it really is important to recognize the asymmetry of Einstein's equation. So you can have whatever opinion you want about matter and the geometry of spacetime. But you're not allowed to have the opinion that the TAB field determines the metric. It just doesn't. It's just a mathematical fact. But the other direction does hold. You give me a metric, it determines a TAB field. So there can be no disputing that fact. Yeah. You mean just there is a freedom about vacuum. So when there is no matter, you have different options. Yes. But it's not just the vacuum, though. So do you think that there's a mathematical theorem that says, you give me a TAB field that's non-vacuum? I can construct a unique metric from that. Because I do not know of that theorem. Who proved that theorem? No, I'm not a specialist in such theorems. But what I'm saying, as soon as you have matter, you cannot have a flat solution. Well, that is true. Yeah. Okay. We can call this a weak determination of matter. Okay. As questions, debates. Okay. Thank you for the talk. You guys have combined different works in it. So with this, we should finish. Someone has the last remark, but it's already quite late. So thank you all for being here and the speakers and the respondents and the public. We will continue these discussions and some further questions. Thank you so much. Thank you very much. Yeah. Thank you.