 Mynd i'r ceisio, mae'n hyffords i'n lluniaeth o ddim yn mynd i'n gaveyn, a at ydych chi'n ddweud bod mae'n dweud bod sicrhau ein syml yn cael eu rwylaid. Felly dri ennyddw Ieidwch am yr ysgol y Pracopa Theorem, y bydd y Brun Mynkowskij, oherwydd eich ddigon nhw ynghent yn bwytoi'r cyfnod pethau a'r hyn sy'n ei maen nhw. Mae'n rhaid i'r gilyt pau ydw i, neu rhaid i'r gyrch yn fwrdd nesaf, I will ignore issues about when this interval is finite or infinite. They are assuming every convex, a complex theorem says that Jesus comes to the many theorems of the form you start off with some convexity operation to get something else. Switch are nm, right, so I will never forgive you, David, for this. So that's what we are going to have to get. This is a well-known in convex analysis. Sometimes this is referred to as the functional version of Bromankowski. So let me also say why that's sort of true. So there is this advisor, Bromankowski theorem, which says the following. So K sitting aside, say, R in the Messiah was convex, the same convex body. And then you can take out this here, this is the direction and then this was K. So I think the slice is Kx. So the slice of X I call this Kx. The Bromankowski says that the map X goes to minus the volume. So I do minus the volume, it's minus the volume in front. And why are these two things essentially the same? So this Procopter implies the Bromankowski. So P implies the n by taking the support function. So I think F to be the function of zero X by taking K. So if K is convex, this is convex in a generalized sense. You see that you're picking up exactly the volume of the fibers here. That's essentially what Bromankowski says. It also implies a minimum principle. I should not write that once, because that's what we've come up with today. It also implies a minimum principle. Which is if I take the infeamon of this modern Fx Y, of Y. Some think it's a full game. Is it less general than the power of one of the M? For this? Yes, I mean less general. This was the discussion that was just happening. Yes. I mean some... It's weaker. But is it weaker? Yeah. For Bromankowski, this is equivalent to that other statement. Or it turned out to be better. The other statement is that Bromankowski... I'll treat this way for the other statement. I'm not thinking that it's going to happen. I've no idea that if you take this marginal function as convex. So this is all in the real setting. This is actually very easy to prove directly. This is so much simpler. But I decided to put it there for two remarks. The first is that there is a version of this in the complex setting. Because we saw a little bit in David's talk. And this is a complex for a couple of complex Bromankowski and Eamon. First one. Again, there is a minimum principle. It's just... How do we get involved in this? So for many years we discussed in the presenters... We're on the heated short-flow. Over time, I realised that this was a very flexible thing. It used very little of the theory of potential theory. The only thing it used was the minimum principle. In a talk, and I just said this, I said this really is a very flexible thing. It should just be totally general. It should be much more general than what I'm talking about. But I have no idea really what I meant. And not even at the talk, at the dinner. I think you said that dinners are important too. At the dinner, the host said to me, You know about this sort of completely generalised complexity called F-sapharmonicity. And F-sapharmonicity, so right now this is exactly a much bigger framework that we call these... Many of these ideas really pull through in a couple of years. Can you repeat the term? Yes, F-sapharmonicity. F-sapharmonicity is exactly what I'm talking about. The idea is to say what I mean by generalising these things and then to generalise something called F-sapharmonicity. And this is following Harvey Paulson, but also many other people as well. There is a framework based on what other people are going to try to say. So what is F-sapharmonicity? So the idea is this. That would be to move by P instead of N by N on symmetric positive definite matrices. Symmetric. It's very simple, which is if you have F-sapharons are smooth, the smooth functions is just quite the same point that convexity is. And the idea of F-sapharmonicity is just to replace this set P with some other set of symmetric matrices. You really do get a theory with very great role in those type of policies, right? So let me make a definition so I can say something precise. So if you wish there's that. Symmetric matrices. My matrices are proper, close sets, and the neocrotiny is such that F-sapharmonicity is right. So you take any one of these matrices, you add on a positive matrix here. The basics of the theory that's really the only thing you need and this is really a very flexible theory. And it's flexible in the sense that there are many examples of such apps, not just arbitrary examples, but interesting examples, before there's a potential difference. Okay, so maybe I should add that. I've got to tell myself let's just say what it means for a function to be subharmonic. So a smooth app with functions, I guess is quite clear what you might want for a theory here. Usually a potential theory you don't want to deal with smooth functions, you deal with up-and-down continuous functions. I'd say that G is up-and-down continuous. You can deal with minus of the G's here. Okay, so you might not have a world of five estimates. So what you can do with it is you can say that if you test it to a point x0, let's consider a function F is smooth and lies above it. So I say that G is F-sapharmonic. I guess more points move away. For all the smooth F, this will go this way out. Yeah, you do. A function that is minus infinity will then test it. Can I say that? I just test it at a point where it makes it. So I don't have that example. I mean, okay, but let it do a much simpler example. Let's just take what happens, right? Right? So it goes away from the corner of the edge, right? That's how it's not an issue. And then at the corner, there are just no such test functions. Okay, so then you're probably right. Okay. So that's how it works. So I test it at a point. So, again, let's do some examples. So, for example, if F is d, I'm just getting back to that. You can do that with this. I've seen that the second example I'm going to have is F is f-sapharmonic. F-sapharmonic is set of all the second. Anybody makes these. If it's positive, it's a negative, they're not f-sapharmonic. It is a sort of the two main extreme examples, the most natural examples, but we can have many other kinds of sets. F-sapharmonic takes time to discuss even. Too many of them, but let's just think about what these two things are. Let's say we do the case N is 2, so we have 2x2 matrices, and you look at the spectrums. We look at the eigenvalues of these matrices. So, of course, the positive matrices are the ones whose spectrum lies with both the eigenvalues and then the sub is where the eigenvalues are, or the trace, so some of the eigenvalues are positive and I think it is positive. But if you see, you know, these are two convex sets, I can do anything that's sort of a convex set in between. So, just another example, it's sigma in Rn, it's a symmetric property that you add on anything positive, so if you add sigma, then I can let S sigma be the set of all A symmetric matrices whose spectrum lies inside the spectrum of A and then, again, you get some notion of a convex set that's interpolating between these two. So, again, trying to be efficient, let's just talk about what that theory is. So, even with this sort of very flexible definition, a lot of the things that you know are still true, like, for instance, F-harmonicity that's done under uniform limits, under point-wise decreasing limits, you can take envelopes or such things. I think some of the theory that you know, and I think one of the sort of highlights of the work of Harvey Lawson is that they show that the Dirichlet problem can be solved for this F-harmonicity and I'm sweeping a lot of their work under the carpet to do something that needs to define what it means to be F-harmonic, and then the Dirichlet problem makes sense that I disagree with it. So, I didn't want to talk about that, I want to do a precooper theorem and you imagine I'm going to say one of those and next I'm going to say, you know, with this theorem there's precooper theorem, that's correct, right? But I need to be a little more careful because when I've done this for this F-harmonicity, what I've done is I've just defined a notion of subharmonic functions that define, functions of find anbargals, right? For precooper I need something that's also defined of N plus anbargals, right? So, before I can say something, let me make a definition, i ddigon i fynd i'r ystyried ychydig sydd wedi'i gweithio y prosesedd cyffreddau cylliddydd. ti chi'n fryd do i'n dwi'n dweud. Brныwn arall oedd o'n mynd i ddim yn ei gweithio'n byw. Roeddwch chi'n dweud a'r cyfl gefnogi, mae'n gyfwyl i gydig o'i ddim yn rhoi'r cy科ol ddweud, mae rhaid i'ch gweithio'r ddigon i'n roi'r cyffreddau, cael ei gweithio ar hyn oherwydd ymlir ychydig i'r rhaid i ddweud, Fy hwn. Wyth dim yn y lleol eraill, ond gallwn dda i'r newid y Maesffordi passenger yw'r cyllid. Felly yw'r cychwyn raiseun gyda hwn, yn ôl y cymdeithas, ac yn amlwg bod fy maesfa yn y cyllid gyda'r cyllid yma, ac yn eistedd yn ddefnyddiad y byw yna, a wnaeth angen fydd wedi'n ein lleol yna, am y gychwyn gyda'r cymdeithas yma Chonfrannol, ac rydyn ni'n fwy fydden ni'n ddechrau. a gallwn am 1 Aではo ychydig yn ddiddor ei amd�dol, a gennym yn ddiddor ddiddor i'r ddiddor i'r ddiddor. Gael eich parwydau a dweud mor bod S-R-P sy'n teimlo wedi cael ei ddiddor. Wedyn ond yn ddiddor y bynnag mae'r gwneud sut mae'n bimlach yn ei ddiddor. Mae ddiddor ddiddor wedi gwneud ddiddor a ddiddor yn ddiddor. Mae ddiddor yn ddiddor fydd yn ddiddor ei wneud ddiddor. I will in this second. Yes? Since I haven't said anything, the answer is no, but maybe I've done something, that I have said before, is you don't understand what I have said. I've heard it from a lot of people, sy'n ddigwethaf yn oed, rydyn ni'n ddweud o'r unrhyw yn gawb your word. Rydyn ni ddweud allan gyda gwaith ar y cyfyrdd. Rydyn ni'n teimlo gan fy m lac o'r cyfyrdd. Rydyn ni'n ddweud ar ddweud o'r ffaith. O'r gwaith rydyn ni'n ddweud i gael gael gael, a ddegwch! Ond yn meddwl i'r pwrdd am gweithio oherwydd o ran nusgofau o'r llun. O'r ddweud o'r gweithio oherwydd o'r llun oed wedi gael o'r glas, No, mae'r rhaid mewn bydd yn fuddwch wedi'i ddweud ein bod eich casio dynnwys. There is no easy function for fzef ornoddau, yna'r halau. So nid ydych yn cynnig hefyd mewn dynnwys procopa, mae'r fzef o wneud yn hyfryd o bwysig mae'n bwysig fel fzef o ddynnu. Mae hwn o'r fzef o bwysig yna'r hwyr ddynnwys spearad. So, when FSP is exactly Procopasterum, and otherwise it's something else. 2022, so this is the version of Procopasterum. As I said, it really contains the original one that says all of it. Okay, so we need two more. Procopter, we also have this generalised problem in Karsby. So, again, let's have K city of Sardare, or Sardare, and let's suppose assume K is... So, here I'll pause for a second. So, there are many ways of characterising what a convex A is. So, here I'll just say what I mean by this. There is this some smooth function of neighbourhood of K, that is FSP subordinate, and so that K is the separate row as we consider it. So, things get as long as you consider it. So, this is what I'm going to mean by this asset being convex. So, again, if F is P, this is just the usual convexity, and then Brooming Karsby holds. So, there are things to say. Are there any questions? Every soon, I will invite the minimum principle and the anti-mixture to have a minimum principle, so F. If this is the closest, I have to answer your question about the PTs. So, I mean, I think from convex F, there is kind of an interpretation of what it means for this F subordinate C to be, right? And then this is... Yeah. Yeah, so it's a pretty abstract, I think. I mean, maybe the way to think about it is when F is... Yeah, sorry, I could say that. It's all other PTs, right? So, when F is P, right, then homonicity becomes the real one-champere equation being zero. And when F is F sub, it becomes, as you say, the low-passion. And in between is something in between. And then you can also see that I can't really like how it works, but with F being... If F is, say, cut out by, let's say, to begin with a finite number of hyperplanes, then for each of those, we get some sort of PD and some sort of... ...collectual first. Yeah, sorry. Right, so I've been discussing only the real things. So you can, of course, do this all with the complex test. Right, so let's just say that a complex to which they set would be the same conditions made on the complex Hessian, right? And then you would place E, is there a positive definite, a real matrix is a positive definite, a mission matrices, and then you get pruysau homonicity, exactly. And we would say, we would do this directly with the outbreaks, that's how it would be the same, right? So, so... Generally, I remember like 25 years ago, everybody was using, like, a month's drill-off for complex mojapur, but I never seen explanation why is complex mojapur a special case of those re-operators. I only had to, like, re... I know you had to prove some, but obscure paper that came down could be quite exciting, quite a lot. It goes very, very quickly. Sorry. Now, you may be thinking about something. The set-up works in the complex case for the complex Hessian, I've not yet tried to make this work. Here I'm only thinking, for real, there's probably a complex version as well, but we'll cover your work. We haven't done that yet. There's got to be some slightly changed statement, because you don't know. But there has to be something, right? A complex version of a complex theorem has to involve some other hypothesis, like David was talking about, independence of the imaginary domain. So, something extra would have to be done before I just say this rule works over the complex. Sorry, David. Maybe that's related to your definition of the historic peak complexity. Here's how much complexity we have. That's probably different from here. Here's how much complexity we have, and this is the level of definition of complexity. Sure. But it's probably different from here. No, I think then there's the set. Okay, so putting this aside, just this side, if I replace P with positive definite complex, the Hermitian matrices, then I get Pluris upon honesty. Yes. And I get this statement as well. Pluris upon honesty, I get no changes. Right. Here probably everything holds true with a little bit of a lot of the one-actuality policies that just happen. So, it's about time for us to come here. Yeah, of course, to come here. We need to have a picture. This would be matters for me. Yeah, CACM, complex apai. I mean, if you get that, it's not going to be. But that's in the face of the explanation why the critical pie is not true with the complex that is sure we use any sort of use. Okay, so what do I have? I'll take the rest of the time. Okay, so let me just bring this back to maybe things that David, like interpolation. So you can do interpolation of sets and you can simply do interpolation of complex functions. So let's go back to interpolation of sets. Discuss two interpolations. I guess the first being the complex interpolation and the second one being the harmonic interpolation. They both fit into this. If I could let A be the smallest of omega, then this will actually be the original matter data. So let's assume that omega has some sort of complexity. Exactly, strictly. Presumably everything is complex in the second variable. This really is a family of complex sets. This A will be complex in the slices. So for example, if F is B, the complex interpolation is meant to be something you can just see. Maybe it's a proposition. We prove that the harmonic interpolation that David was discussing is the smallest set in this case. You can take this F sub. Of course, you've got many more, right? For any given F, you have some other interpolation. We're interested in this. So we will answer now an email to say something that is expected. Time will be the set. Future idea, something expected. I may have to say something that is expected as well, which is that it's not just a couple of months but I expect that there's a whole bunch of other pieces of theory of complex geometry that should come as well. One thing that I was thinking about when I came over here, if you've got an email like that, is the idea of the next volume. This is a statement that we make here that is to remind you of the next volume. There are two complex sets. There is an R2. R2, I can take the volume of, say, the sum of A0 and A1 by some non-negative numbers. It turns out that this has a polynomial expansion and it's provisions of the next volume. So I just look at T0 squared times B0 squared. And the final number is V of AI and I. V of AI. These quantities, right? Of course, if you put T1 is 0, I'm just scaling A0, so this is just the volume of A0. 1. And then this is the third volume. You have three mid volumes. You have two sets, right? You have three mid volumes. You can pick the first set twice. The second set twice, you can pick the two different ones. It's six or eight things. This has to do with two for the two volumes. There's a two. The main one up here, right? OK, so... there's almost no interpretation of set. If I continue to find me that I have two sets, what does it mean to have the next model? So I don't know for sure. Let's just think about it for a second. I think again it comes up with this idea that if you want to... and I think of this as a... like an interpretation like David was saying. this case right so this case A4 I should we do be given A0 and A1 as being two sets over zero one yeahуки teblog zero one that one of the continuous functions on the boundary, right the boundary of course that says two points there is just two functions and it says the delta zero and the delta five okay so what you are doing with selecting A 0, A1 is youre just saying if im going to delta zero twice im im going to delta one twice im im going to take delta zero and delta one just do that in the same thing with this stroke So ddim ei boddych yn ddechrau eich hyffordd, dwi'n meddwl stwyd, dwi'n meddwl iddyn nhw i'r llwer i ddo i relatedad, lle allanodd mor identifiedd holl. Felly roeddwn i', roedd yn meddwl i'r holl yw'r holl yw rhaid beth o'r holl, dwi'n meddwl i'r holl, dwi'n meddwl i'r holl mewn holl. Yn mynd i'r holl y dathlen dwy'n cofyrdd mor fforsあれw'r mynd yn i'n meddwl. Felly roedd yn meddwl i'r holl ar y plwyd. ddysgwys, ar gyfer sy'n gyflwyno. Rydych chi'n gweithio i ychydig o'r cyflwyno homonych. Ikrwys ychydig i bod yng Nghymru'r Gweithredu, fel rydych chi'n meddwl i ddiffinogu hwnnw o geitio'r cyfrifysgau. Mae hynny nad yw e-A Quandoff bwysigach yr ysgrifet. Oherwydd i'r bwysig ymghefnidd o'r gwybinton o'r gwybwyno, Fe oedd gwneud dros eich cwylig, a rwyf yn gwneud y peth i'r bobl yn lle i gweithio'r bobl arall Yonganon, Felly mae'r dabelwch yn wahanol, ond rwy'n gwneud ond mae'n gweithio'r bobl a phes Wait%, ac mae'r ddigwaraeth.