 So it's a great pleasure for me to introduce Peter Schultz who will talk about the joint work with Dustin Closen and new foundations of real functional analysis and the title of his talk is liquid R vector spaces. So Peter, that's yours. Yes, thanks a lot. Thanks a lot for the invitation to speak here. So yeah, great pleasure to speak about my joint work with Dustin Closen. So in particular, I want to speak here about the stuff about what we did over the real numbers. So the basic question, I mean, we somewhat proposed these kind of new foundations for doing topology to work with condensed set instead. And as Dustin explained, one key point is that condensed sets are also really nice from an algebraic point of view. They basically like an algebraic theory. And I mean, also going back to Lovari, that was a suggestion to use topos to study functional analysis. We defined this chronological topos, I think, and proposed this as a possible new way to do functional analysis. And we have slightly different topos called condensed topos. And we really claim that in this condensed topos, you can really do it. So the question somewhat is can one use the condensed topos? Sorry, who suggested the topos? Maybe it was Johnstone? Johnstone? Actually, not confused. That one was Lovari and Johnson had this topological topos. Sorry. I think Johnson brought the paper on a topological topos. So yeah, there have been previous proposals for stuff like that. Condensed topos to somehow, sometimes algebra is not functional. So what do we want roughly? We want a nice category that we call liquid object spaces. So these should be a full subcategory of the condensed ones. And so what you should have someone in mind as the analogy is that some of all condensed object spaces, it's roughly analogous to some of all topological object spaces. But if you work with all of them, then you can't really control those. In particular, like if you try to form a tensor product, then you do want to take some kind of completed tensor products when you work with topological vector space. You don't just want some kind of abstract algebraic tensor product. And so one standard category that's usually isolated on when one works topological vector spaces is this class of complete locally convex object spaces. That's like the standard class of topological vector spaces where all the examples like Banner, Persie and whatnot spaces live. But it's also a class that one can reasonably control. And so roughly these liquid object spaces, they should be something like the complete locally convex. That's the hope. But in particular, there should be some kind of completion function here. It should be well behaved in some ways. We also want a nice tensor product. Let me just say that this is computable in practice. Actually, this is tying to the early work of Grotendieg to his late work in some sense. I mean, his early work was exactly about functionalism, exactly about the problem of defining tensor products of Banner spaces. And it seems like this. We define this projective tensor product, the injective tensor product, many intermediate ones. And it's also right here. So what are some of the easy router that we would like to have here? So we definitely we want to keep the good categorical properties. Men's topic spaces are like generally, I mean, all this condensed stuff is very good categorical properties. And we wanted this class of complete ones. So in particular, you want that it's still in the being category, so that one can do homological algebra in this setting without any troubles. We want that it's still generated by compact projectors. And maybe this is, I mean, actually, the existence of compact projectors would actually be formal once you have this kind of left edge on completion time here, because this is compact project generators. So if you want this kind of feature, this would be automatic. There's a B and it's not quite automatic. And maybe it's not so clear that one should hope for all of these properties, but it will be true. So maybe it's not something you desire on Ferrari, but it turns out that you can get it, and then it's very nice to have it. So this is getting it should also be stable under all limits, the co limits, and extensions. But before I want to get into this over the real numbers, I want to briefly recall a similar thing that you can do a non-accommodian function on this over the periodic numbers. So interlude the analog over the periodic numbers. So P is some fixed prime number now, QP. So you can just as well consider the QP vector spaces. And then we define this notion of a so-called solid QP vector spaces inside of all components. And let me briefly recall how solid it's defined. So there's some completion, this recall solidification, of course. And so, okay, so here, let's first try to understand what the compact projectives are. So as Dustin explained these categories that we consider, they are very often generated by compact projective objects, and really those are the ones you have to understand first. And here it's a very general as a compact projectives for any condensed ring, the modules would be just the QP of John S where S is one of these extremely disconnected profiled sets. So one of these compact projective condensed sets. These are, the problem is that these things are rather inexplicit. There are some of the elements there are just formal sums, formal finite sums of points in S with QP coefficients. And then you give it some kind of funny condensed structure. But the three guys here, the three solid QP vector space. Well, let me just say what it is, it's exactly the space of measures on S with QP coefficients. So one way to define this would be to take the internal home from the continuous functions from S into QP. So yeah, so what is the measure really? And it's something that to any continuous functions associates a number. So yeah, that's what it measures would be. A slightly different way to write it is that it's also, you can also, if you write, if you write S as an inverse limit of finite sets SIs, it's a finite, so it's a profiled set, right? So you can write this as protected from the finite sets. Then dense subspace of the continuous functions is really given by the characteristic functions of open and closed subsets or the QP linear span of these. So it's enough to say, to define a measure, it's enough to say what's the measure of each open and closed subset is, and each open and closed subset somehow comes by a pullback from one of these finite quotients. So there should be a way to describe the measures somewhere here in terms of subscription. And what you get in this way is that you take the free guy on SI, takes the inverse limit of all of those, so this and some compact to be in group, and then you just invert P at the end. And so in general, the condensed QP vector space V is solid, if, I mean, that can be taken as a definition, but only for all such profiled sets S and all maps from S and to V. Well, what should it mean to be solid? Well, it means that, so whenever you map S and use a free guy on S maps in, right? That's what it means, the free solid QP vector space on S. So this here maps to the space of measures on S with QP coefficients. And there should be a unique extension where this map here, how does S map into the space of measures, that's just by the Dirac measures. So coming back somewhat to the general picture here, like, if you don't pause any completeness, if you look at all condensed QP vector spaces, and you can also think of QP drawn as some space of measures, but it's just a space of measures that's just a finite atomic measure, right? Just to find some of Dirac measures. But if you want to encode some kind of completeness, then you should ask that certain infinite sums are also convergent. So when you have some certain measures supported on infinite sets, or even like some kind of harm measures thing that supported on the whole profile set in some sense, then you should be able to integrate a lot against any such measure. And let's go get something. So let's have Tilde here. Do you get some extension here? And the way to think of this extension is that if you evaluate this have Tilde on some measure mu, to mu some element in the space of measures, this is the integral of f against this measure, okay? So this already looks a little bit like function noises. So that's how we can define these solid QP vector spaces. So you ask that whenever you map some profile in seven, or just to make sure you disconnect the one that doesn't matter in the end, you can even really map the whole space of measures, okay? And so from this characterization, I mean, you can actually, if you start with a Banach space or a Frichet space, then it's quite easy to see that these kinds of integrations here against measures can be defined. And so the corresponding condenses QP vector space for such Banach or Frichet spaces, they are solid. So Banach or the QP or QP Banach. Well, it's containing QP Frichet. These are containing solid QP vector spaces. And the really nice thing is, I mean, you do get some kind of solid tensor product here. And the really nice thing is that these inclusions, they are compatible with tensor products. Okay, so you can define tensor product here. I mean, it just comes in such a way that some of the completion is compatible with the tensor product. So one way to compute is someone to take the tensor product, condense QP vector spaces and then re-solidify. And it turns out, and that's a quite non-trivial computation actually, that if you do that to QP Frichet spaces, then the tensor product is still a QP Frichet space and is a usual tensor product. And so this way you can really embed QP Frichet analysis into the setting of solid QP vector spaces. And what you gain here is really that this is in a BN category and you have a really nice ambient derived category. And if you want to do all sorts of homologic algebra, it's really convenient. It has been used now in some papers on Katie Cox theory, just formalism by Dito Bosco in particular. Okay, so what we would like to achieve is the analogous picture over the real numbers. The goal is to achieve the analogous thing. But there is some kind of very important technical problem I was trying to do this. So recall that that the whole condense formalism is very much based on profaned sets. Condense sets are glued from profaned sets. And QP is of course locally profaned. ZP in particular is profaned. It's counter set. And so to this, these speak to each other. But the real numbers, they're connected, they're continued, they're not at all profaned. But the condense formalism always asks you to somewhat chop up your real number, chop up everything into some of profaned sets. So you need to chop up your real numbers, cut them again and again and again into small and smaller counter sets, and reassemble some of them in the end. This seems like a very strange thing to do about the real numbers. And it's not at all clear that you can really control this. But R is decidedly not. I mean, some of the real technical problems one runs into is to how to resolve nice arvector spaces by the three condense arvector spaces on profaned sets. And if you want to compute some x-quip, you do have to use these projective resolutions and say condense R modules. And the projective optics are joined as various with some extremely connected set. And so if you want to compute some x-quip, you have to resolve by these guys. And absolutely, you know, you can do it. You can always find these rejections from these extremely connected sets. But the way you do this is completely inexplicit. I mean, use these complexifications of some of discretized vector spaces that you can operate, and I hope to get an explicit control on this. And so, yeah, someone not so clear how you can really achieve that. Okay, so that's the problem you will run into. But that's first, just try to naively redo all of the real numbers, what we did with the periodic numbers. So here's the first naive try. First try. Maybe not completely naive. First try for a series of liquid vector spaces. Now, by the way, the reason this is called liquid and not solid, those real numbers is that there is this kind of very general notion of a solid being group, so it's defined with the integers, and which are like QP, some are specialized for solid QP vector spaces. But the real numbers are not solid. And if you have a condensed real vector space, it is solid, it must be zero. So there's no solidness, it's somewhat way too strong to work over the real numbers. And so we need to do something else over the real numbers. So if you think of all condensed things as probably clouds and profile sets as something gaseous, then the solid things are like the most cooled down things, most rigid things. And these liquid vector spaces will define as somewhat intermediate. They're not as not as, I don't know, like a gase, they're more condensed, like liquid, but and sometimes they will even depend on an exterior parameter that might think of the temperature. But you can't make them completely solid. Anyways, that's just the names. Okay, so what's the first try for a theory of liquid vector spaces? So we want to define the three complete guy, profile sets as our extremely disconnected one. Turns out that usually practice, you can always find every single profile sets already, not just for the extremely disconnected ones, even if they do play an important technical role. And then we already can practice it. Yes, for what one should do is to take the space of measures on S. So what is this? So one way to think about this is again, just the internal home from the continuous functions from S into R. So this is the Banach space was a sort of particular to logical object space. And so you can regard as condensed object space. You could also simply define this as internal home from S into R. This also has immediately a condensed structure, which is the one you would think about. And then you can then suspect R and take the internal home. And so this is some kind of space. Several different names, I think for this. So sometimes they're called science, but all measures, it's just the usual space of measures that's considered in functional analysis. So any compact host of space, even you can define space of measures. And then maybe there's some discussion about which topology put to put on this. And well, there's one answer to the party, which would agree with the science structure. So basically, it's a weak topology. It's compactly generated version to be very precise. And so actually, that's a way to write this. That's very analogous to how we've written the space of measures of the periodic numbers here. And then what you can do is you can, first of all, I mean, for each of these finite sets as I, you can take the three vector space on SI. And I mean, for a finite set, okay, so it's like the space of measures on a finite set is R join SI, of course, generated by the Dirac measures. But you shouldn't take the whole inverse limit because this would be way too big. You want some kind of boundedness. And one way to say this is that we always take the part where the L1 norm is bounded by something. If you want to say that it's to bounded measures, then you should ask that like the value on any subset is bounded. And this precisely means that you balance the L1 norm in this vector space. If you say, the L1 norm is therefore equal to some constant. And then in the end, you take the unit here over all constants at the beginning of zero. So again, this space here, and this here is some bounded close subset of some finite dimensional real vector space. So that's a compact positive space. And then by the theorem of Tuchonov's inverse limit, there's still a compact positive space. So this is again some kind of compact positive. Now, of course, this time it's not profaned. It was profaned over the periodic numbers. And then in the end, just scale it out. The analog of inverting P in the other description. So over, this is analogous to what we've done here, except that here, we didn't take the L1 norm, but we chose the L infinity norm. And that's just because over the periodic numbers, we have the ultrametric inequality. So if you say that the measure is bounded, then it's enough to know that it's bounded by one. It's enough to know it's bounded by one and all the atoms, because the sums don't make anything larger. So here we somehow did the analogous thing by using the L infinity norm, sub-norm, and then take an inverse limit and then inverting P is the same as union. But if you want to have really an inverse system here, I mean, it's really necessary to take the L1 norm here. Otherwise, there wouldn't be anything right now. Right. So that's what we would try to do. And then there is a basic proposition that if V is a complete locally convex topological armature space, and you have some map from S and to V, this is pro-planet, and then there exists the unique F tilde from the space of measures to say the corresponding convincing vector space. Right. And so this looks encouraging, that like taking these things here as your basic built compact projective folding blocks might give you a good theory. So in particular, as a banner for space space, it's the only thing I'll satisfy for this property. So that's a good thing. But then you run to a problem. And the problem is that the X1 of such a space of measures against R can be non-zero. So for example, S being the one time computation of the integers. Okay. And so that's the problem because, so there exists such an extension, here you have the space of measures, here you have some, some extension V, and here's the wheels. And I mean, here you have S nothing in it by the rock measures. And actually this lifts because the obstruction to lifting is given by some H1 of S with real coefficients, and this turns out to vanish. But then if there was this nice category that's stable under extensions, then this F should extend to a splitting. That's the X group is non-zero in general. So it follows that if you want to category to be stable under extensions, then this cannot work. And if you think more about it, you can also show that it's whatever you want to write down, it can't be in a BN category. It's slightly easier to someone show that it can't be stable under extensions, but it also is true that it's not. And the origin of these extensions as, I mean, this paper from Rieber, he gave an example of such an extension. Maybe not exactly for the space of measures here, but for the closely related space of L1 functions on S, they are very, very close. And this extension is also defined as R space. Such extensions can be constructed from the entropy function of the Shannon entropy. The H of some sequence of numbers XI, this is some of the XI times log XI. And to keep on about entropy is that it's locally linear, locally almost linear. So the Shannon entropy of some XI plus the Shannon entropy of some YI minus the Shannon entropy of some XI plus YI is bounded by a constant times. Well, let's say the sum of the XI is normalized to be one in z greater or equal to zero, and this is equal to constant. And maybe to make sense of this, I should divide this by two here. Or generally, if you have a sum of scale, A times one plus one minus A times the other, then it's bounded by a constant, but not globally almost linear. And it turns out that somehow what is X1 group here measures in some sense locally almost linear things, because if you want to do some kind of extension of vector spaces, you can write down kind of locally first, and then you need to write down a core cycle that's almost linear. But to define a splitting, you would need to somehow global almost linear. It needs to be globally almost linear. So that put us down for quite a while. So what this shows is that V is not locally convex. This extension V is not locally convex. So it turns out that this notion of local convexity is just not a notion over the real number set stable under extensions. There's basically no way to force it to be so. So it is just strictly necessary to allow non-local convex space into the picture. Okay, and so then we learned about this space, this word autonomous, etc. So there's this notion of P convexity. And now suddenly P is not a prime anymore, sorry. Where P can be any number between zero and one. So P Banach is like a Banach, some complete norm vector space. But the scaling behavior is different. The norm of A times V is the norm of A to the P times norm of B, where P is V. I mean, I guess they're different ways of defining of saying what's normal. So I wanted the norm still satisfies the usual triangle in the folding, but the scaling behavior is different. And so some example of this is it takes the kind of the LP space on the integers. So this is a set of all x i, i, n, such that some of the x i to the P. And this thing here, which would send these a norm. And I mean, sometimes one again, that takes a piece root of it to get the usual scaling behavior, but some other triangle inequality, but it's a coincidence. I find this expression slightly nicer to work with it, again, taking the piece root of it in the end. So maybe I should draw a picture. So it's dimension two. So if you have a standard basis element, well, let's start with L infinity. So that's here L infinity, this cube. This here is L two. This here is L one. So I probably can't look at this. And then it becomes non-convex. So then you're in the other half. And then becomes sooner and sooner. And so yeah, we really need to look at these non-convex figures here. So and so there's a theorem of Calton that an extension of P banners is at least P prime banners for P prime less than P. So I should say maybe say that P banners, if you have a P banners, if you have something that's P convex for some P, there's also P prime banners for P prime that are at most P. And I guess for this, I need to rescale my norm, but it will still satisfy the right term. And so yeah, you might lose a little bit of this convexity when you take extensions, but just infinitesimally stole P prime banner for all P prime less than P. So in particular, if you don't ask that it's like one convex, but only that it's less than one convex, so P convex for all P less than one, then this is a condition that has a chance of being stable under extensions and maybe also higher extensions and so on. And this is actually what we go for. So so here's the definition. And everything from now on will depend on an exterior parameter P that lies between zero and one. And the fact that the theory of liquid object space somehow depends on an exterior parameter is I think the profound mystery of nature that we don't understand yet. Okay. So fix such a parameter, then three P liquid object space on some profile set S is the space of less than P measures on this. So let's define this union of all P prime less than P. So P prime measures on this, whereas these P prime measures on this are defined just in the same way as the measures just replacing the L1 norm by the LP prime norm. It takes a free guy on these SI's and then you take the part where the prime norm is at most. And then we can define what a condensed, when a condensed object space is liquid or P liquid is for S and S functions into D. Well, again, there's this map here into the Dirac measures in the space of all measures you want to take this unique extension. Okay. So then the theorem is that this defines an analytic ring structure. In particular, so we didn't say what these analytic rings are, but in particular, P liquid object spaces are in a being category generated by these contact projectives and S and P or S is extremely disconnected. They're stable under all limits, cold limits, and extensions. They have a nice symmetric normal tensor product. There's a slew of structure. I mean, you can also have a nice derived category. There's a whole slew of structures that you get automatically once you have such a normal structure. Right. So again, so some of the banner spaces, they're contained in the free space spaces, they're contained in the P-liquid ones. And the one slightly annoying thing is that the tensor product here will not in general restrict to the tensor product you have here. But actually, you can't really expect that anyways, because Groten-Dick defined not one, but several tensor products here. So sometimes you wouldn't even know which one you should get. And maybe that would be a natural guess for which one should get, but anyways. But what about the nuclear case, the case when- Yeah, that's what I was just about to say. So inside here, you have the nuclear, I appreciate, which are the ones that actually most often come up, say when you consider smooth functions on a manifold or holomorphic functions and so on, they're always nuclear. And on them, it gets in the correct tensor product. So that's again, a quite controversial computation, but it works out in the end. And so, yeah, so you get this beautiful ambient category, very nice to be incredibly general by compact projectors with tensor product and so on. And in practice, you can't compute the tensor products, because most often you're dealing with some kind of nuclear free space space. So in it, in Dustin's, Dustin is currently giving a course about the Rammco homology. And so, particularly explains this term of Groten-Dick, that if you have an algebraic variety with the complex numbers, and it's the Rammco homology degrees, it's algebraic Rammco homology degrees, but it's not really the Rammco homology. And I mean, the key thing you have to prove there is somehow about, when you have boundaries, normal crossings, you have to prove some kind of funny version of a concrete lemma that's some kind of analytic and algebraic version of the same homology. But using this liquid formalism, this immediately reduces to the one variable case, because in general, it's just a tensor product of several, this thing and several variables. And so, with no work, you can reduce to the one variable case, but it's just a really, really simple computation. So, I don't know. So, I think having such formalism can really be quite helpful. Right. So, this theorem, a reconjector to theorem, maybe, I don't know, I think in January 2019, and then we basically spent all of 2019 trying to prove it. And this was really quite a nightmare. So, the proof is pretty hard. But there's one search that we reduced it to rather early on. So, the key assertion is the following, key lemma. So, let's say this is a P-banoch and P prime is less than P. Then you need to know that the x-groups of this space of P prime measures, and that's against V0, V5. So, this is precisely saying that this phenomenon that we could have this, where do I have it? All right. I mean, the problem here was that we could have these extensions of these space of measures against the real numbers here. And, well, you don't want the extension against the real numbers, but more generally, you don't want any extensions against P-banos. And yeah. So, if you would have non-trivial examples, for example, then again, you would run into trouble for the same reason as here. So, you definitely want it all such extensions against the P-banos. But actually, to really get this whole nice category of analytic grid structure, you don't just need to mention of the x1, but you need all the higher x to manage. But the problem with proving this is that if you want to do this, I mean, you have to compute some x-group here. And so, how do you compute x-groups and condensed R vector spaces? Well, you have to resolve by the free condensed R vector spaces on compact projective sets. So, you have to, but this forces you to resolve R vector spaces by profiling sets. It's just not really possible. And so, for this reason, and also because we are both number-series and we always want to have some series that define not just real numbers, but somewhere elsemetically, we decided that we will instead try to prove something more general, we opt for a generalization. Namely, you can define a certain ring of, let's fix some radius between zero and one. Then you can define a certain ring of first method for normal series, D-larone series TR is a set of all those normal series, some a n times G to the n, such that the sum of the typical way of a n times R to the n is finite. So, if you want, you can regard this here as the union of C of some part that is less than or equal to C. And so, this is naturally a profiled set. Because what happens here? I mean, when you put such a bound here, it follows that each individual coefficient a n can take at most finitely many values because otherwise this contribution alone would be too big. And it must actually be zero for large enough n. And so, then you just get a product of finite sets or a closed subset of the product of finite sets for this thing here. So, it's a profiled set. And we have a suggestion from, if zero is less than or prime is less than R, then this space of C series just surjects onto the reals by evaluating it R prime. And so, this presents the real numbers as a quotient of this arithmetic ring by some one element corresponding to the set prime. And so, what we do is actually we so we can also define space of measures over this finite arithmetic ring. Again, by taking the union of all C greater than zero of the inverse limit wall i of the free guy on S i, where again, you just pass to them, that's equal to C sub space, that's defined in the same way. Actually, I mean, in this sense, it's actually some kind of L1 norm that you really take here. And then there's a funny proposition that this once can very canonical space of measures that you can define integrally, and that's somehow, you didn't have a parameter here on some kind of LP norm parameters, it's just using the L1 norm on the integers. And if you take that, and then model it by this one element here to get something with the real numbers, and this canonical is the same thing as a space of P measures on the reals where P is taken so that R is R prime to the P. And so, this thing interpolates all spaces of P measures. So if you're somehow very, very, the R prime in this interval here, so you have so lots of different real points of this thing, which is basically some kind of curve, it's basically a principle that you'll remain the same, and it has lots of different points. And at these different points, you get these different spaces of P measures on the real line. So there's really some real varying family of copies of the real numbers with these different spaces of measures. And so when you let P go to zero in some sense, this space of P measures, then in some sense, you enter the arithmetic region of the screen, so it really seems inextricably linked. The arithmetic is really inextricably linked to what happens with the real numbers. And okay, so what really happens here is that so if you take R prime to be a tenth, then we're somehow really thinking of real numbers here in terms of decimal expansions, right? So you can write any real number as some a n times one over 10 to the n, if it just takes a decimal expansion. But decimal expansions per se, they are somehow naturally from a co-finance set because they are built from finite sets zero to nine. And so in this way, we write the real numbers here using something like decimal expansion in terms of something locally pro-finance. And so now if we try to, by generalizing our results, if we want to prove those real numbers to this arithmetic LaRom series ring, we overcome the problem that the real numbers are not locally profiled because this ring here is locally profiled, each of these subsets is profiled. So we can state analog over the real numbers. Sorry, over the LaRom series T bar, and this is locally profiled or it's a union of profiles. So I mean, they can hope to find such a project of resolution that you can still understand. Okay, so that's the first step. And I think it's a rather beard step still because we're trying to really just set up foundations for real functional analysis. And the way we do it is by doing arithmetic really work with the integers. And another key thing happens, you regain some discretized convexity. So because we had to prove the statement about these P-banner spaces, so they are not locally convex. So both of these files are some non-convex vector space. And this just means that many of the techniques that you would like to do, they don't really work. But after you go to this arithmetic LaRom series ring, then here we're really taking the L1 norm. So this is really behaving like something convex, except that it's somewhat discretized. So you can't always take the midpoint between something, but if there is some of something, some net point that's close to the midpoint, then it will still be somewhat the right norm. So yeah, so we also regain some convexity in doing this discretization. And okay, so yeah, we were able to push it through. I mean, so we use this Brine ring resolution. And again, that's slightly weird because we want to do an explicit calculation, but the Brine ring resolution is not explicit. So it seems like a mismatch, but it has enough nice structural properties that you can somehow even do it, although it's not explicit. And actually, one key structure that we use is the homotopy between, well, what does that use the net argument is that multiplying by two internally and externally on the Brine ring resolution is somewhat the same. Here we need to use something slightly stronger than that, namely that addition internally and addition externally homotopic on the Brine ring. So that's the kind of structure you have. And then, yeah, at some point you somewhat need to use that here, this p prime is less than the p that you have here. So that you somewhat, the norms on the two sides somewhat have different skating behaviors. That's somewhat the key thing that here someone's in here. To multiply by a real number, then it has a different effect on this side that it has on this side. And this on the end, you're able to prove the result you want. But, yeah, I found the argument quite nasty. And so I made this challenge to the computer formalization community to verify it on a computer. And it turns out that they were able to do it. And just a few weeks ago, they completely formalized the key technical theorem that I was unsure about. So yeah, so it seems to work. Thanks a lot, Peter.