In this video, we're going to be going over a center tap by phase rectifier. So as you've seen in our other videos, a rectifier takes AC and converts it into a pulsating DC. So we've got these humps here. We're going to take a look to see what happens to my lovely humps. What happens is, if this is negative and positive, let's just say that this is negative. It's going to go through here. It blocks there. It goes across my load, which gives me a hump that way, but on its way back, it gets on its way back here, sorry, it gets blocked. So you'll see that gets blocked. But then on the second phase, as we see, we've got negative here. So current's going to flow through here and go there. And as it goes back through, it's going to get blocked. So then we just get that one hump and that side's blocked. Then it happens again, away we go. Current goes that way, but on its way back, it gets blocked. So we get this hump, and then one more time. I think you're picking up what I'm putting down, goes through, but gets blocked. So we end up with these humps. So we get this pulsating DC going across here. No more AC for us. So to see how this really works, we're just going to assign some values. We're going to try to figure out what our peak voltage is at the load, what our DC voltage is at the load, what our peak current will be at the load, what our DC current will be at the load, the power being dissipated across the load, and what our peak inverse voltage needs to be for these diodes. Now let's just start seeing what we can do with the voltages here. I've got 1200 volts across, so I'm going to have 600 there and 600 there. So we've got 600 volts across each winding. I've got a winding to winding ratio, if you remember, from transformers of five to one. So if I have 600 here, that means I'm going to have 120 volts over on those windings on my lower voltage windings. So therefore, I have 120 volts that I'm dealing with here. Now that's a center tap by phase, which means that the AC, if these guys weren't here, this guy would see 120 volts in one phase, this guy would see 120 volts on another phase. So we've got a three wire center tap. Now in all these questions, the first thing I want to do is I'm going to take my RMS value, which is 120, and I'm going to drive that up to peak. 120 divided by 0.707 gives me 170 volts peak, which coincidentally we wanted to find out. So there we're going to put it over here, 170 volts peak, and we're on to the next. Now that we have 170 volts peak here, we can determine what our DC value is by taking our peak value times 0.637 to get our DC value. Remember this peak times 0.637 to get DC. So I end up with 108 volts DC across the load resistor. So we get that drawn in there. Our next thing is to work out what our current peak is going to be. In order to figure out what our current is, I had to assign a resist value. In this case, we're going to use 10 ohms of resistance. And if I take my peak value of 170 volts peak, divide that by the 10 ohms resistance, I end up with 17 amps peak, which we're going to put over on this side now. 17 amps peak. Next, we've got to take this to a DC value. Well, just like we did with the voltage, we're going to take 17 amps peak, and we're going to times that by 0.637 to get my DC current value, which in this case is going to be 10.8 amps DC. So we're going to put that over on our side over here. We have 10.8 amps DC. We're getting close to the end. Power. We need to figure out what our power is. So the power being dissipated, we have a voltage here, and we have current here. We can determine what the power being dissipated across this resistor is. In order to do that, we have to remember the formula. E peak times I peak divided by 2 equals power. Well, I have my peak voltage here, 170 volts peak. I have my peak current of 17 amps peak. So I just go 170 times 17 divided by 2 is the average power being dissipated across this resistor. And in this case, that works out to be 1,445 watts. Last thing to figure out is we need to figure out what the peak inverse voltage, what's the maximum voltage that this guy has to be at least to block. Now in this case, with a center tap by phase, normally I have just 170 volts across here. But if you take a look here, this diode here has to block not only that center tap, it has to block the entire voltage across here. So its peak inverse voltage is not just going to be 170 volts. It's going to be 170 for this coil and 170 for that coil because these two coils are in series. So these diodes need to be able to block the entire thing, which is 340 volts peak. So the peak inverse voltage of the diodes on a biphase rectified circuit is two times the peak value of the one winding or 340 volts. And that's it. That's how you take the steps you need to calculate everything you need to know in a biphase rectified circuit.


Okay, so just continuing our little tour of Edinburgh. So I don't know if you guys realize that Scotland is part of the United Kingdom, but it's continually threatening to break away. So what the Royal Family did is this is the Britannia, the Royal Yacht. When they retired it, they put it in Edinburgh as a sign of solidarity with Scotland. And then in another way that you show solidarity, they built a huge mall right next to it. So this is interesting. It's on an inlet from the bay and the boat's parked there and you can go through it. But there's this massive mall you have to walk through to get to it. So it was an era, it was a redevelopment project to redevelop the downtrodden parts of Edinburgh. So I figured what the heck, you don't get to go on the Queen's Yacht very often. So this is the Britannia and it was the Queen's Yacht starting in 1953. And this is it.  And I mean, it's important for the Queen to get away from her duties in the palace.  And so it's important to get away on occasion on your yacht. And so this is the yacht, really nice, you know, wood decks, beautiful yacht. And this is the rest of it on the side. And I mean, this is a chip. I mean, this isn't just, you know, even Trump doesn't have one this big. And so this is a nice shot.  And so this is the formal dining room that's there. This is a little taxpayer fund. And this is, of course, I mean, England, the taxpayers always fund the monarchy. This is the sitting room, very, very nice. What was amazing to me, though, is the bedrooms. So you've got this palatial yacht. So I said, wow, this is going to be really spectacular. And then the bedrooms where the queen sleeps is about the size of two of these desks.  That's it. That's it. No, no, not even a queen bed.  I was just stunned.  And so this is the sitting area, of course, with its fireplace, and it's got the windows all over. Now, my favorite is you have to have an operating room on there in case something happens with the queen. So this is the OR, you know, circa 1950s. And so, or the VA, you know, this is the VA OR. And so this is the OR, and you've got your anesthesia machine there. And there's a surgeon on board.   And so she's got her own OR here on the yacht. OK, let's get back to real work here. So our final real path lecture today is going to be on tumors. And when I'm starting on tumors, you can't really just talk about childhood tumors without talking about the differential diagnosis of leukocorea. So first of all, Rhys, what does leukocorea mean? White. White pupil, literally, for much language? Degree. From the Greek, of course, from the Greek.  So you've got a young child with a white pupil. What's your differential diagnosis? What's the first thing you want to worry about? Retinoblastoma. Retinoblastoma, OK. Cataract. Cataract. Coats. Asymmetric refracting. Oh, OK. On a list of 10, that would be 15. No. OK. Albinism. Again, that would be 14 on a list of 10. Astrocytocamartoma, that'd be like 10, maybe. Exactly. We used to call it PHPV. Now it's called, you may see that written in place. Now it's called persistent fetal vasculature. ROP. Retinopathy prematurity. Yeah, well, that was, we already said that, yeah? Well, yeah, you always want to, exactly. End up the Midas. Always keep that on the list. All right, we cover them up.  OK, so, all right, so, yeah, exactly. Toxoposmosis, that's infectious, so good. All right, so back to Rhys. So you're looking at a young child, say this child's like two and a half years old, and you die at the pupil, and you see this. What do you see in here? So elevation of the retina, just behind the lens. Oh, sorry, wrong button. All right, I'll figure out where we get to, here we go. OK, so you've got, it looks like there's, these are retinal vessels, and so that tells you maybe something's behind the retina, pushing it forward, because if you look, you see that the retina looks like it's pushed right behind the lens. And so if you look inside, and now you see this. What are you seeing when you look in here? Calcium. All right, so you see the little white. The little white areas here.  So does calcium help you when you're looking at your differential diagnosis? Indeed it does, because if you're looking for a retinoblastoma, retinoblastoma is calcify, whereas the other lesions that cause bucochlorida do not. And so one way you can tell the difference is you can do a B-scan ultrasound. And remember, the ultrasound, when it hits calcium, will bounce back real strong, so you get these strong spikes. Now, problem is, is two-year-olds don't let you do ultrasounds. So you have to do all this while they're asleep. But if you do an ultrasound, and then slowly turn down the gain, where the calcium is, it'll be all lit up, and then the rest of the aisle disappear. And so retinoblastomas calcify, whereas other potential causes of bucochlorida usually do not. Now here we see an eye that has been cut in half sagittally. And this is a large retinoblastoma. And again, if you look, it looks white. And you can see it's coming underneath the retina. Now, Eileen, what are the two growth patterns that you can see with retinoblastomas? And no fitting with exophytics. Okay, and what are those, how are they defined? Exactly. So if you look right here, you can see that the retina is draped over the top of that and pushed forward and the tumor is growing underneath. So this would be an exophytics. Exophytic means exo outside. It means it's growing outside the retina or underneath it, if you will. Endo is in, so endophytic means it's growing into the vitreous. And so you can see a large tumor here. Again, see the chunks of calcium all through this tumor. Again, a good way to tell that it's a retinoblastoma. And then here's another one, same thing. You can see it growing under the retina and then this is all exudate right here. So this is some exudate that you get from the tumor itself. All right, so Adam, this is showing you a tumor at lower power. And what can we make out on this tumor lower power? What are some of the things we can see when we look at low power? Exactly. So these little right kind of magenta colored areas are calcium on an H&E state. What's the growth pattern here? So it's mostly endophytic, but if you look, there's a couple of areas here that are growing under the retina. So you can have both. You can have both endophytic and exophytic. And this is mainly endophytic. It's growing into the vitreous. And then you can see a little bit growing exophytic. All right, so what is the characteristic findings that you see here at Medium Power? We'll go to Chris, right? Chris, okay. What am I pointing out here? What's the difference in these areas here as opposed to those areas there? A lot of areas kind of more global or I think it's very much about the same. Gosh, I was gonna see if I could get you, maybe a third of that is right. So if you were a baseball player, you'd be worth $10 million now because that means you're successful a third of the time. Okay, so what this denotes here is the darker cells are viable cells. These are cells that are still viable. But if you look, each one of these clusters of viable cells has a blood vessel in the middle of it. So there's a blood vessel here in the middle of each of these clusters and then there's some cells around it. This is actually not stromal connective tissue. This is dead tissue.  And so basically what you're seeing here is viable tumor cells around the blood vessel. And then you're seeing areas here where it's just necrotic. These pink areas are necrotic. And what happens when this necrotic tissue dies off, it can become secondarily calcified. They call this dystrophic calcification. So that's why you see calcification in here.  And here's a close up showing you that. Here's some viable tumor cells here in these clusters. And then in between this pink area this is dead tumor cells they've outgrown their blood supply and they've secondarily calcified. So dystrophic calcification. All right Ashley, what feature is this that's commonly seen in these tumors? Okay, so home or right rosettes you can see in neuroblastomas. But in retinoblastomas there's a more specific rosette that you see. Julian? They're winter steiner. Exactly, flexner wintersteiner rosettes. There's the Germans that say flexner wintersteiner. So flexner wintersteiner rosettes. And these are rosettes that are characteristic of differentiation of retinoblastoma. So when you see these it means the tumors differentiated. And you see it's got these round clusters of cells. And the reason that that happens is that these tumors arise from primitive retinoblasts, primitive retinal tumors. And so when primitive retina goes bad it starts to make round structures instead of flat structures. And so these little rosettes, it's funny if you look at them, you could almost get a hint that it's trying to make little rods and cones out here. I mean, if you really hallucinate with me sometimes you can even see a little outer limiting membrane here. And so these are attempting to form retin except retina usually doesn't have one of these structures. Ashley, what is that? Mytotic. Mytotic figures. They usually don't have that in normal retina. So, and there's a nice mytotic figure there. So these are the flexner wintersteiner rosettes. These are what we see as a sign of differentiation in these tumors. Now for bonus points, let's see that kind of counted. It's all right, you get a count on that. Renee, what kind of stain is this and why would I be showing you this? Okay. And you're trying to stain for mucopolysaccharides. Very good, extra brownie points for that one. So this is an ocean blue stain which if you remember from your corneal dystrophy stains form mucopolysaccharide. And if you look inside here you can see the mucopolysaccharide usually the stuff that's underneath the retina, you know where the rod and cones actually are between the RPE. Now I used to think that that's what kept it stuck down there, but it's not. It's negative pressure pulling from the coroid. It doesn't really get stuck down to the RPE. And so there is mucopolysaccharide in the center of these flexner wintersteiner rosettes. All right, Jack, what are we showing here? I mean, here's some rosettes here. That is this thing. Okay, so they're kind of palisading. You see they haven't shown a circle here. They're almost like a C-shaped. They're almost like a circle that's becoming unzippered. What is that called? Nick. All right, this is called a floret. And it is a sign of even more differentiation. So when you see florets, that means that the tumor is even more differentiated than a rosette. So it's like a rosette unzipping. And so that's a sign that the tumor's really well differentiated when you see this nice, pretty floret. I guess back to Rhys. What are we showing right here? What part of the eye are we in? And why would I show you this? So it looks like iris and lens on the left side of the screen. Okay, iris, lens here. And then the vitreous behind that, all the way to the right, just filled with tumor cells. Okay. What is this right here? I mean, I think I would call that vitreous seeding. Exactly, this is vitreous seeding. And so this is kind of a funny cut. It's way in the periphery. So it's almost like a flat section. These are little ciliary processes. And so you can see tumor cells are broken off into the anterior vitreous. They're behind the lens. They're even in between the ciliary processes here, all the way up behind the iris. And so this is vitreous seeding. Is that important? It is. Why? In terms of treatment, you know, you either have to do the vitreal chemo or take the eye. Okay, yeah. So that is important because it's tough to get treatment into there. And so when you do the intra-arterial chemo, it works pretty well nowadays. But sometimes if you've got vitreous seeding, it doesn't work too well. People are even advocating, injecting chemo directly into the vitreous itself to try to get rid of these vitreous seeds. And so this is a sign that the tumor is spreading. You don't like to see cells in the anterior vitreous because that's a poorer prognostic factor. All right, what are we looking at right here, Eileen? What's wrong with this picture? Blue cells, what else? Well, there's definitely tumor cells in the angle, which is a bad sign, but this picture is upside down. I do this every once in a while just to see if you're paying attention. So I'll throw in an upside down one every couple of lectures. So this is upside down. This is corny up here. This is iris back here. This is tumor seeding into the angle. And so when you get a retinoblastoma where the tumor cells have broken off, you can often get seeding in the angle.  We've even seen severe retinoblastomas where it's been misdiagnosed as an endothelitis because you'll see these kids with these white, fluffy cells actually layering out in the anterior chamber angle. And it's just seeding of the retinoblastoma cells. So here you can see these retinoblastoma cells have gone into the anterior chamber. They're even in the angle and in the trabecular mesh work. Is that a bad thing prognostically? Yeah, because it can spread. And so that's a bad place for tumor cells to be. So you can get not only seeding of the vitreous, but seeding in the anterior chamber and even pseudohypopions from this. All right, Adam, what's the worst prognostic factor when you look for retinoblastoma? Exactly. So when these tumors spread, they spread by direct invasion into the optic nerve. So here you see tumor here and look, there's tumor into the optic nerve. And so this is an old, old slide I borrowed from the AFIP. This was probably, this slide's probably from the 60s sometime. And this is actually the, you know, percent of kids that die depending on where the tumor is. So this is before we had, you know, chemotherapy, intra arterial and intra vitriol and all that. And so the bottom line is, is if you have tumor just inside the eye, this was at that point, you had like an 8% chance of dying. But if it goes through the, I'm gonna crybrosa, it doubles. If it goes into the nerve itself, but not at the cut end of the nerve, it goes up even higher. And the worst prognosis is at that distal end of the nerve, if there is still tumor there, that means the tumor can get into the orbit and even go back to the brain. And so the worst prognostic factor is tumor at the distal end of the nerve. And so when we get a retinoblastoma specimen in our lab, when boopy or whoever's taking these out takes them out to take, cause a lot of a piece of nerve as they possibly can, first of all. And the first thing we do is we cut the distal end of the nerve and make sure that the distal nerve is free of tumor. Cause that's important prognostically. So if you still have tumor at the distal end of the nerve, you know, they had a 65% chance of dying 50 years ago. All right, and this is what happens when you don't treat it. And so this is a child from Nigeria. And unfortunately retinoblastoma is very aggressive. And so if you don't treat it, you know, in the third world, sometimes these kids are not, you know, have any access to medical care. This tumor can take over an entire orbit. So this was a retinoblastoma.  All right, so we're looking at other possible entities now that, first off, any questions on retinoblastomas? Now realize that on OCAPs, OCAPs have a lag time of a few years because treatments have to become widespread before they can quiz you on them. And so I don't know if there's gonna be any questions on intra arterial or intra vitriol chemotherapy cause I don't know if it's accepted enough and the greater communities show up on OCAPs. So I'm not sure, you know, if you need to know the specifics of that or not. Sometimes it's a lag time. So questions? All right, so now we're gonna look at other entities that can cause leukocorrhea. Chris, what do we see in right here? What's funny about this picture? Okay, what makes you say it would be Coates disease? Some areas are normal retinoblastoma. So you see those funny, beady, dilated, telangiectatic vessels, that's a little tip-off.  This is actually a kissing exudative retinal detachment, if you will. Believe it or not, this little cavity here, that's what's left of the vitriol. And there's so much exudate under the retina that it's pushed it forward. So you got a total exudative detachment of the retina. The reason it's yellow in color is it's the lipid-rich exudate underneath there. And these are all these bizarre, tealangiectatic vessels. All right, so tell me, what's the usual age range in the Coates? Yeah, so older than a retinoblastoma, certainly. So six to 10, six to 12, unilateral or bilateral? Unilateral, good. Okay, so again, I'll tell you what I told all these guys. If you're making a guess, you say it with conviction.  So they'll look at it while you're guessing. You say unilateral. So even if you're wrong, you still, at least, you get credit if you were right. So just guess with conviction. So unilateral, excellent. And male or female? Male. Yeah, so more male. More males. All right, so here again, this is another view we're looking inside. You've got a little normal-looking retina here. Here, this retina is elevated because there's all this yellow lipid-rich exudate under the retina. Here's all these telangiectatic, bulbous-looking vessels that cause this. When you look, here's an eye that was actually nucleated due to Coates disease.  And so it's like, you know, the state food of Utah, you know, jello. The only difference is it's not green, but it's very gelatinous. If it were green, it would be Utah's state food. And so it's very, very gelatinous. When you cut these in the lab, it literally feels like jello in there because it's a lipid-rich exudate. And here is a retina. Here is one of these telangiectatic vessels and these little empty vacuoles here are dissolved lipid. And so these little telangiectatic vessels leak this lipid-rich exudate and eventually it can gather under the retina and give you that total RD. All right, so actually, you know, Ashley, we'll save that for you. Anybody? What are these banana-shaped white areas here underneath the retina? What do they signify? Very good cholesterol. And so for some reason, cholesterol, when it gathers in an exudative detachment, it'll form little banana shapes. I don't know why, but sometimes they'll be straight and long for another time. Look at that, forms a perfect banana there. And so the cholesterol will gather in clumps and then it'll dissolve on our tissues process. So this is lipid-rich. These are little macrophages all over the place, stuffed with this lipid, trying to chew it up, but they get overwhelmed. They just can't keep up with it. So Coates disease. Ashley, what are we seeing right here? Persistence. Okay, and what makes you say that? Good, so you see this whitish mass, and if you look carefully, there's what's left of the crystalline lens right there. So this mass is either behind the lens or maybe even invading into the lens.  So this would be consistent with persistent fetal vasculature. We used to call it PHPV. Primary hyper-persistent, primary hyperplastic vitreous. All right, so if you see the term PHPV, that's what it's referring to. So again, age, actually, no, no. So older than retinoblastoma. So it's not usually recognized until a little bit older, but not as old as Coates, so kind of in between. Unilateral or bilateral? Unilateral. Unilateral, exactly. And I don't think there's a sex predisposition. Not that I recall. All right, so this is an eye that's been cut, you know, cut kind of tangentially from behind, and this shows you that when this lesion grows into the lens or behind the lens, it tends to pull the ciliary processes in. So it almost pulls them to the center when you look at it. All right, and Julia, what is this showing? So this is the persistent high load artery coming in, and you can see this mass taking over almost half of the crystalline lens, and so oftentimes they'll grow behind the lens, but sometimes they'll actually grow into the lens, and you can see right here where you've got that high load artery there, and then going here, and look, here's the ciliary processes being pulled to that. So this is actually the same eye that we took that picture of at the very first, showing you that it's invading half of the lens and then pulling the ciliary processes in. So it's there when they're warm, but it doesn't get white. It doesn't get white until, I don't know why, it let's proliferates, and you almost get connective tissue growing in there, and blood growing in there, and so it takes time for it to grow on. All right, so this is an eye that's been removed, and you can see the problem here is you have a complete stock of that high load system coming from the optic nerve head, back behind the lens, and then involving them. But you look right here, the retina just isn't well developed in that in general. These are tough to treat because you think, oh, you just go ahead and take out the lens and do a vitrectomy, and you can, but the problem is the retina's just not well developed, and so these kids just don't do well, even if you do a heroic surgery. So here we have a crystalline lens. Here's the stock of the persistent high load artery, and here you can see multiple little vessels here, and then this is growing into the lens, causing scarring and opacity of the lens. All right, here you see a nice view. Look at that ciliary process. It's like somebody grabbed it and just yanked it in. So there's that so-called elongated ciliary process growing in here. Now, this is a kid. This just shows how our legal system works in this country. This family was really concerned. Dr. Harry did an ultrasound on this kid, about three different people looked at him. They said, man, this looks really scary. We think maybe this is retinoblastoma. There was some iffy changes on there. It wasn't clear what it was. And so the decision was made to enucleate the kid. And so we looked at the path and we said, wow, this is just really nasty persistent fetal vascular, not retinoblastoma. And so the doctors involved went to the family and said, you know, we've got great news for you. This turned out not to be a tumor. This was something else. So the kid doesn't have a chance of dying down the road. This is great news, at which time the family sued them. Wow, you removed an eye that didn't have a tumor in it. So this is our legal system here. That's how it works. And so you take a news and say, great news, no tumor. Oh, well, we're suing you then. And so they wouldn't have sued it, there really been a tumor there. But they're suing because they removed the eye that didn't have a tumor, even though the eye had no visual potential. So welcome to the legal system you're going into in this country. So this is that kid. And you can see they had this fibrin reaction in here. They had a kind of a pseudo-hypopion. And they had this big growth back here. The scleric process is yanked forward. And there again, you can see that this is the aurasirata. Even the peripheral retina is being yanked into that. So it just pulls everything to the center when you've got the PHVV. All right, Renee, what is this? Okay, so this is a less significant remnant of the hyalurid artery. So instead of having a full-blown persistent fetal vasculature, this is just the posterior remnant of that hyalurid artery. So you'll often see a corkscrew vessel. Any clinical significance to that? Absolutely not. So you just tell the patient, yeah, you've got a curiosity in the eye. It's not gonna cause you any problems. Don't worry about it. So that's just the posterior hyalurid that hasn't quite regressed. What's the equivalent of the anterior hyalurid that hasn't quite regressed? Okay, so this is that little remnant hyalurid artery. Sometimes you can see that in the vitreous. You'll see a cloquage canal where it runs. And oftentimes where it attaches here, you'll see a little dot on the posterior surface of the lens. I'm sorry, I don't have a good picture of a mitendorf dot. You see them all the time. We just don't think about taking pictures of them. And here's that posterior corkscrew vessel coming up here. And that really has no visual significance. It really doesn't cause any problems. All right, Jack, what are we seeing here? All right, so does that change your kind of thinking on a differential if you see leukocorrhea bilateral? Okay, so you can't have bilateral Arbis. So don't forget that. How about Coates disease or persistent fetal vasculature? So short-term memory, we just said that. They're unilateral usually. They're usually unilateral. So what's a bilateral if it's not a retinoblastoma? What are other causes of bilateral leukocorrhea? Okay, those are usually unilateral. What's the cause of a bilateral? Cataracts, good. That's one that could cause you bilateral leukocorrhea. What's another that could cause you bilateral? Something we still see a little bit nowadays, retinopathy of prematurity. So retinopathy of prematurity can give you bilateral leukocorrhea. And this is what it looks like when it's resolved. This is called a drag-to-disc. So what happens is when kids are born prematurely, that temporal retina hasn't finished vascularizing yet.  Basically, then you'll get all kinds of vessels starting to grow in that non-perfused peripheral retina. Those will eventually start to contract. You can get attraction, retinal detachment, and all kinds of problems. It was interesting in the 1950s and early 60s, they discovered that when kids were born prematurely, if you gave them oxygen, you could save them. Except one of the side effects of too much oxygen is you would get retinopathy of prematurity. And there was a huge epidemic of ROP at that time. And then as people got better at treating these kids and realizing you don't give them too much oxygen, you monitor it very carefully. The incidence of retinopathy of prematurity dropped. But now, in the last couple of decades, it's creeped up again. And the reason for that is now we are saving kids who would die previously. I mean, these kids are coming out now. These kids that are like this big, they're like little fetuses, and we find a way to save them. And so the more premature a child is, the more chance they could develop retinopathy of prematurity, even with our modern technology of following their oxygen level. So this is a so-called drag disc that we see in this resolve retinopathy of prematurity. Now, if we don't treat retinopathy of prematurity, then you can get a total exudative type retinal detachment. And they used to call this retrolental fibroplasia because it just means you just get this kind of white mass behind the lens, and it's a total RD. So again, these are really difficult to treat once they get to this stage. And so we can prevent them. You guys do need to know this for O-caps. You know, we used to do cryotherapy in that non-previews peripheral retina in these little neonates, but now people started to do indirect laser. And there's less side effects, less inflammation with that. So I think now that pretty much the party line, I think most people here just do the laser rather than the cryo. But if you can laser or cryo that area of non-prefuse retina, you could prevent those abnormal blood vessels from growing and prevent these total retinal detachments, so-called retrolental fibroplasia. Okay, so for a bonus point here, Nick, this is a kid that had eye removed for severe retrolental fibroplasia. What kind of surgery has he had to try to save the sign? Sclerobuckle, exactly. So you see, there's the buckle. There's the buckle. So they did a sclerobuckle to try to save the sign. You do have a trachomy, sclerobuckle. Again, these are very difficult to treat once you get that stage five closed funnel. Extremely difficult to treat. All right. What do we got here? Kind of a weird-looking eye. We're kind of going pretty far down on our list of causes of leukocorina now. I mean, so there's kind of an exudative retinal detachment. Yeah, so an exudative detachment, kind of a white mass right here, so kind of a focal white mass. What could give you a focal mass, causing a white pupil and an RD-like look in a kit? This is the path. Toxocara, so there's actually a beast in there that's been broken down. I mean, this is a broken down beastie. So toxocara, which unfortunately comes from puppy poop, basically, and what do puppies do?  Then what do they do?   And so what happens is the toxocara gets into the system that goes through the intestinal wall, then it can spread to end organs. And one of the end organs is the eye.  And so if you can find a focal area, you may want to try to go in there and do a trick to me and get it out of there, but these are, again, difficult to treat. Okay, Eileen, boy, extra bonus points. Now we're really going down to obscure causes of leukocoria. What part of the eye are we looking at here and what the heck are we looking at? Ciliary body, there's the iris, ciliary body. What kind of tissue is this? Actually, it's a normal tissue in an abnormal place. This would almost be like a chorostoma. They love this stuff on board. It's just really obscure my new show like this. Believe it or not, this is Highland Cartilage. So you need to remember that. What entity gives you Highland Cartilage in the ciliary body? Well, actually it could. That's a good answer, but what they look for in these is trisomy 13. So trisomy 13 can give you Highland Cartilage in the ciliary body and can give you leukocoria. And this is the other thing that occurs with trisomy 13. Now this is not a retinoblastoma, but you see those rosettes? Malformed retina can form rosettes. And so in trisomy 13, you get in the ciliary body Highland Cartilage and you get these dysplastic looking rosettes in the retina. Now trisomy 13 is even in this era is pretty fail. I mean, these kids don't live very long and so not a good thing to have. So trisomy 13, obscure cause of leukocoria, but if you see Highland Cartilage in the ciliary body, jump all over that. All right, so now we're gonna go to more tumors of adults. And of course when we're talking about tumors in adults, the one thing we always talk about is pigmented lesions. So Adam, what do we see in here? Okay, are there any features of that that are alarming? All right, so it's big, it's kind of thick. Looks elevated, good. Now fortunately it's not going into the angle. You can still see a clear iris there. It's not quite pulling that pupil, but this is a non-dilated look. So it is kind of pulling that pupil over there a little bit, a little ovelling of the pupil. So you would be concerned. And so whenever we see pigmented lesions like this, our concern is, is this a nevus versus a melanoma? Now is this more concerning? It is, so look at the slit beam. Here's the beam on the cornea. Here's the beam on the iris. Look, that's almost touching the cornea. It's markedly elevated and it's going into the angle. So you would be very concerned about this particular lesion for melanoma. So when we spent time in the lab, what you guys will do is you guys will read up on the diagnoses, the various causes of pigmented lesions of the iris. And when I was a resident, it was taught that iris melanomas were different than melanomas elsewhere. They have a very good prognosis, you know, 90% prognosis because you can see them and you treat them earlier. Well, Fred Jacobiak, at that time he was in New York, he looked at about 180 of these at the AFIP and said, wait a minute, let's look at these. So he took 180 quote iris melanomas from the AFIP, the Armed Forces Institute of Pathology which doesn't even exist anymore but was the repository of tens of thousands of tumor cells for 100 years. So I think you can still get ahold of the tissue if you want to do studies. But he looked at these and what he found is the reason that the prognosis is so good is that 80% of these are really just nevi, not melanomas. And so he came up with a classification of iris pigmented lesions and it's real important that you know that. So Adam, this is the first one that we look at. What do you see in here? Mostly spindle cells. And no, there's no nucleolionin, there's no pleomorphism, they look really benign. So what do we call this? Spindle nevus. So the first benign lesion that you want to know about in these quote iris melanomas, spindle nevus. Chris, what do we see in here that's different than that previous one we showed? Man, you've got good eyes if you can call those mitotic because I can't from here. But what I wanted to show you is look, there's tumor cells in the iris stroma but look, there's like a whole bunch of them on the surface of the iris. That's a plaque. So you can also get spindle nevus with a plaque and those can look pretty scary when you look at them clinically. So spindle nevus, spindle nevus with plaque. All right, Ashley? There's still a lot of these little spindle cells here. Look at all of them. But if you look next to them, some of these cells are now becoming rounder or more oval. Not quite epithelioid but more round or almost cigar-shaped starting to show a little bit of features that would concern you. But there's still areas in here where it's these benign spindle-y guys here. Okay, so it's called a borderline spindle nevus. And so these are where they're starting to look a little bit more malignant, starting to become more oval, more cigar-shaped. So borderline spindle nevus. How about this, Renee? So this is actually a spindle melanoma. This would be the equivalent of a spindle-B melanoma of the coroid. So this is actually spindle melanoma. So even just with spindle cells, you've got a couple of benign, a borderline, and then finally, this is a spindle melanoma. So the equivalent of like a spindle-B melanoma of the coroid. All right, Jack, what are we seeing here? I'll give you a hint. This was bleached because this was so heavily pigmented you couldn't even see the cell details. So we actually had Mary bleach these. I've got a ton of melanoma. So this is what's called an epithelioid nevus. These are rare as hence to. So I think I've seen one of these in my whole career. But you can also have epithelioid nevi. Just remember that. And then finally, Nick. What's going on here? Look at that cell right there. Is that where you? That one there, that one there, that one there. Place here. You don't have to do it too much though, right? You don't have to do that. All right, absolutely. Look at that huge nucleolus in there. Giant nucleus. Actually, outline of the cell border right there. Oh, okay. Outline of the cell border here. So what kind of cells are these? These are epithelioid cells. And so just like a melanoma elsewhere in the coroid, you can get spindle melanomas. You can get mixed spindle and epithelioid. You can actually even get epithelioid-type melanomas of the iris. So those are the ones that we think of as really tumors.  So the key thing you wanna remember is, in iris tumors, you can get similar to coroidal tumors. You can get spindle B. You can get mixed and get epithelioid. This is an epithelioid melanoma. Nucleoli in there, large cells, pleomorphism. These, if you look, we bleached them. You look at the nuclei. The nuclei are small and round. And you don't see activity in there. You just see a bigger cell stuffed with pigment, but you get a smaller on nucleus. This would be an epithelioid nevus. But even a melanocytoma looks like this too. It's kind of a benign, big epithelioid-looking cell. So they can look like this. Epithelioid nevi are, as I said, I've seen one in 28 years. It's very, very rare. But again, something that shows up on there. And so there's, you don't differentiate between spindle A and spindle B, melanoma is on iris. Not on iris. Only in the coroid and the ciliary body. Okay. Rhys, what's going on here? So in the face of the picture, I can tell if that's just where the iris, there's just a clot in the iris. Only the way, or if it's a mass down here. All right, now there's something that tells me this is a mass. So you see this right here. Look at this, right? Oh. Look at that peeking out. So it probably could be like a ciliary body. Exactly. So probably a pigment, a tumor rising from the ciliary body. This is one of my favorite patients, really nice guy. He drives up from Champaign, Illinois to Chicago to see us. And he's a University of Illinois football fan. And he has season tickets. And he could see the scoreboard out of the side the first game, and then it started getting blurry by the third game. And then the final home game, he couldn't see the scoreboard. Not quite. And so he came up because of that. And we looked at him and he did have a central PSC cataract, but he had this tumor pushing from the ciliary body onto the lens and peeking through into the angle here.  Tumors that are that big, because indeed they're growing from the ciliary body. And here's another one showing you the same thing. Here's a tumor poking through from the ciliary body behind into the anterior chamber angle and pushing out from the ciliary body into the lens. And so these people, because these tumors grow behind the iris, they can grow pretty big before you have symptoms. And then they'll push on the lens and they will eventually cause a cataract and the patient will come in. So here's a Goni overview. Look at the tumor back here in the ciliary body. And there it is pushing up into the angle. All right, now Eileen, what is this thing right here? Why would I be showing you this? Sentinel vessel, what does that mean? Exactly, so these are great. You know, a Sentinel is something that's out on the edge there. And so what this does is this points to the tumor. So whenever you see a single isolated solitary vessel right there, you wanna dilate that person real carefully and look behind there because this is a hint that hey, something's going on here in the ciliary body. And sure enough, here's a tumor in the ciliary body, heavily pigmented, pushing against the iris, I mean against the lens, and causing that lens to push over a little bit and then again can cause a cataract. No, this is a different patient. Yeah, I wish I would. I should have said it was. Yeah, yeah, it is, yeah, yeah, it's that patient. These are all melanomas. And so when you look at the ciliary body and the coroid, those are all treated the same and we'll go over the cell types of those. Just remember the iris is a little different. Ciliary body and coroid are the same when we classify these. So here you see a tumor, here's the iris root, here's the trabecular mesh work, and here's the tumor growing from the ciliary body into the trabecular mesh work. And so this again is a bad place because what do you have coming out of the trabecular mesh work? You've got slump's canal, you've got aqueous veins. Tumors can get out of the eye in this area. So this is a poor prognostic factor in having a tumor in the ciliary body just by definition. And here you see another one, pretty big one here, growing right here. And Adam, why would I be showing you this picture? Look up here. So sure enough, this is a closeup of that. If you look carefully, here's the tumor in the ciliary body. There's that first vessel we looked at. This is out on the epi-sclera. So indeed, when you get tumors inside the eye that are melanomas, how do they spread? Through like veins or emissarial channels. Through anywhere you've got something coming into or going out of the sclera, i.e. a vein going out, an artery and nerve coming in, tumors, the melanomas get out that way. So retinoblastomas go out from the optic nerve, melanomas go out from emissarial channels. And so here you can see, this is one of these aqueous veins and here's the tumor. And sure enough, there's tumor right up there. So this is spread out of the eye and under the epi-sclera space. So again, a very bad prognostic sign. All right, so now we're gonna look at coroidal lesions. Chris, we're looking at one right here. What do we see here? So how would you tell? What are things you look for that would make you alarmed for melanoma rather than anivis? Okay. Well, you can look, but if this is probably not gonna show any, because it's gotta be at least a millimeter and a half, two millimeters raised to show up. What you look for raised, that's good. You look for fluid, you look for lipofucin, orange dye on the surface of these guys. And then of course, size, size and diameter. So this is consistent with anivis. So you just take a picture of it and watch it. Now, would you be more concerned about this? Yeah, because here's the optic nerve. And there, and that's not a focus because this growth is coming from under the retina, pushing the retina forward. And so this is a big growth coming out from the coral, right? And now here's those little lipofucin dots I told you about. Little orangish gatherings here. You can see there's some fluid around this. You can see it's elevated. So you would be concerned that this would be more than just anivis, that this would be a melanoma. So Ashley, what is the particular pattern of growth of these coroidal melanomas that we often see? Mushroom shade. And so if you look right here, brooks membrane has got the elastic layer in it, a couple collagen layers. So those tumors will grow in the coroid. Eventually, they'll stretch brooks, and then it'll break through. But now brooks will still tether, almost like a lasso. And then the tumor cells will grow underneath the retina. So it'll form a mushroom shape still tethered at the base. So very common for coroidal melanomas to form this mushroom shape. Now, this is just a classic appearance of a coroidal melanoma. Now, in terms of the cellularity, now there was a man named Calendron in the 1930s who classified melanomas. Renee, what type is this? This is a spindle-A pattern. So spindle-A pattern, spindle-shaped nucleus, no nucleolus, indistinct cytoplasm. Some people would argue this is just anivis. This may not even be considered a melanoma. So spindle-A, Jack, oh, I already said, all right, spindle-B is the next one. So spindle-B, they tend to get a little bit more oval with a single nucleolus in them, cigar shape, but still indistinct cytoplasm, spindle-B. Jack, what are these? All right, these are epithelioid. So spindle-A, spindle-B, you can have a mixed type, mixed epithelioid and spindle, and then lastly epithelioid.  Look at these. Bizarre-looking, compchromatin, big nucleoli. You can actually see the cellular borders around these. So these are epithelioid, and the further you go toward the epithelioid side, the worse the prognosis. So these epithelioid tumors are bad prognosis. And again, here you can see mostly epithelioid in here, and then there are a few little spindle ones mixed in here. So this is the mixed tumor. So spindle-A, spindle-B, mixed epithelioid. Calinger had another classification. He called it fascicular, which is just a funny-looking spindle-B. And then his last one was necrotic, where at that time they didn't have ultrasound. They didn't have ways of looking at eyes. By the time they'd removed these tumors, it would just be necrotic. But the key you want to remember spindle-A, spindle-B, mixed epithelioid. All right, what are we showing right here? Are those the mesterial vessels? Exactly. This is a vortex vein draining the chloride. And sure enough, there's tumors right there. And where do these tumors metastasize when they do get out of the eye? What organ? The liver. Okay, so how do you remember that? The old saying, beware the yellow man with the glass eye. Think about it. Beware the yellow man. And that almost sounds like something out of Pirates of the Caribbean. So you say, argh at that, argh. Beware the yellow man with the glass eye. So here we see this is a vortex vein. And look, there's the tumor. There's the vein pushed to the side. There's the tumors going through it. And again, it can get out of the eye via the vortex veins. I don't know why they go to the liver, but here's a piece of liver with tumor in it. And they can even go years later. There've been documented cases more than 20 years later. Someone who had a melanoma of the eye and then they die of a metastatic melanoma to the liver. I don't know if they're just quiescent and then suddenly they grow or what, but for some reason they go to the liver. And so very important that you remember that. All right, a couple of other things we can cover in one minute here, okay? So we are back to Reese. What do we see in here? How is this different from the other ones we've shown? Whatever it is, it's a melanotic. Yeah, so it's kind of a melanotic. It's elevated. It is elevated. It's under, you know, in the chloride. It's kind of whitish looking and you don't have that distinct edges. It's almost feathery around there. What if I say it's a 75 year old woman? It could be a man. Okay, what kind of man? Breast cancer. Exactly, so this is easy to remember. You know, breast carcinoma is the most common cause of metastasis to the core aid in women. But as I mentioned previously, lung is catching up because you know, women are now smoking like men. They've come a long way and so they can die like men do too, lung cancer. But still it's breast lung catching up rapidly. So this is an adenocarcinoma of the breast metastatic and those tend to be less distinct, kind of fluffy looking and they can even be multiple because they are metastatic. And so metastatic tumors can occur in the coroid. Breast most common in women. And in men, this is another tumor and in men, lung is most common. Other ones are more rare like, you know, prostate and things like that are much more rare. So remember, breast, lung in women, lung in men, most common. And this is what I love. This is in the mall at the Britannia in Edinburgh. And sure enough, Mr. Nix, home of the Yiddos. All right, so no matter where you are in the world, there's always some Greek guy with a restaurant somewhere. So Mr. Nix. All right, so we wanted to put the quick review of path for the OCAPs as close to OCAPs as we can. So next Tuesday is an OCAP path review. I've got a two hour lecture that I give to medical students and we're just gonna show them in one hour. So please kind of review this stuff so you're ready. It's gonna be show the picture. You got five seconds to say the diagnosis and we're just gonna flash as many pictures as we possibly can in an hour so they'll be kind of fresh on your minds when you take OCAPs. All right, very good. Thank you. You're welcome.


Transparency is really at the core of this whole Paris Agreement and that has several reasons because without transparency you don't know where you are. In red you have that the emission reduction is defined as a reference level up here minus the actual reductions that you have down here. So it's a gap between the actual emissions say for a given period and the reference level for that period. Numbers kind of are biased because there is not a single number. There are many numbers of there and you can create them. We've also heard about how how this science can be manipulated by politics. Politics in quotes but I want to argue that not all politics are bad. Politics is about the world we live in and we need to figure out how to engage between these worlds of politics and science. I think from a multi-level governance perspective one of the most important things is that we need to learn to communicate and to collaborate better across sectors, across levels and across different types of actors. So in terms of transparency and some of the recommendations from our side on that it should be seen as a great opportunity because it builds confidence in legitimacy. Transparency can actually cause initial frustrations because it might need to rethink some of the ways monitoring an assessment has been done but in the longer term it actually enhances quality. Why transparency is very very important for us because we believe that transparency is an essential key to improve the forest governance to protect our remaining forests.


Hello, and welcome to the Grand Line Review, your source for everything one piece. And today we are going to step into a strange, bizarre world based on a single hypothetical, being what if everyone that Luffy actually asked to join the Straw Hearts became a crew member? With the basic rule of our universe being that if Luffy does request a person join the crew, then they must say yes. And not only that, we're also very much going to examine how alternate universe crewmates may have affected the story going forward from that point. Because our hypothetical does change a pretty incredible amount. It's a really good what if, you know, to examine what this series could have been if everyone Luffy actually wants to become a Straw Heart was that agreeable. Before we get into that, I have an even better hypothetical though, which is what if you became a subscriber to the Grand Line Review? Well, I will tell you, you would receive regular one piece content uploaded straight into your YouTube feed. That's what would happen. And I think we should make that a reality, dear viewer, just as we are about to do with our other hypothetical here. And as for an example of what we're doing here today, let's take our piece. According to our rules, when Luffy asked her to join them on their adventure, she would have had to agree instead of staying on the Island of Dragon things. And then bam, we have to account for our piece throughout the rest of the story. However, because our piece is a filler character and does not indeed exist, she will not be here today. Apologies to our piece fans provided there are any. So let's go through one piece with this very simple rule in mind. And right off the bat, we have a pretty, pretty insane change to make because during his childhood, Luffy actually asked both Ace and Sarbo to join the crew. And instead of refusing Luffy, they now have to say yes, meaning that we have our first two official straw hats. And this is going to cause us a few problems later down the line, but we don't have to worry about that for now. But basically then Sarbo's story will still play out more or less the same regardless. Yes, Luffy is now his captain, but he will still need to flee Dawn Island, have his whole fake out death scene in Curremnesia and join the revolutionaries. Don't worry though, he will be back. But for now, the straw hats are left with two members and Ace is now going to wait until Luffy is old enough to set sail. Then through various shenanigans, this duo is going to make it to Shelstown, where Luffy hears of a certain swordsman, an opportunity that the rubber lads simply cannot resist. And neither can Zoro this time, who immediately accepts Luffy's offer instead of playing hard to get. Then Luffy, Zoro and Ace proceed to defeat Captain Morgan and Helmepo. Very important thing I should bring up about Ace though, actually, because he is now a straw hat, he will not become the user of the Merimeranome. And this is because Ace would not have become stranded on Sixus and therefore would not have found the fruit. That's all right, he's still pretty powerful regardless. Next up, we have Nami, a fairly simple situation because as soon as Luffy found out she was a navigator, he asked her to join the crew. But this time Nami just says yes, we don't have to deal with the whole betrayal of Baratier business. Instead, it's much more likely that Nami would just tell Luffy about Arlong and they will make the trip to Kokayashi Village to deal with him regardless. Much cleaner, albeit not quite as dramatically potent, not quite at all actually. But now is where another big change happens because then we land on a strange island and meet the guardian of the forest, Gaimon. And after a touching scene involving empty treasure chests, Luffy demands that Gaimon join the crew. To which this time around, Gaimon is obviously going to say yes, captain. And thus excluding Luffy, we now have our fifth official crew member who is a shrub in a box. But now things proceed quite smoothly into Sierra Village where Luffy, Zoro, Nami, Ace and Gaimon help to save the village from the black cat pirates and Usopp hops aboard. Very easy recruitment here. But all of this work has made us very hungry so we're off to Baratier on the hunt for someone to provide frequent delicious dinners. And what do we find? Well, we find a Sanji, would be the dude's name. And as per the one universal constant of this video, Sanji says yes to Luffy immediately. And if anything, this whole Don Creek situation goes much smoother because the straw hats aren't split up by Nami's betrayal due to their being, no betrayal. Although Zoro still gets pretty messed up by me, Hawk. But with Sanji in tow, we now head to Kokayashi Village where this intriguing incarnation of the straw hats takes down Arlong, frees Nami up. And now we're pretty much ready to head into the ground line. And after doing so, our first stop is Reverse Mountain to acquire yet another crewmate, this time being old man Crocus. After discovering that Crocus was a doctor, Luffy very excitedly offered him the chance to become the doctor of the straw hats. And when I say offered, let's remember that Luffy more often than not demands things rather than offers and his exact wording was something like, you're a doctor, then become our doctor. And in this world, Crocus does need to accept this. So the straw hats now have a ridiculously powerful ally at their side. Laboon does, however, remain at Reverse Mountain, having now become Luffy's rival and will await the future pirate king's return. Also of note is that during Reverse Mountain and the ensuing Whiskey Peak arc, we do pick up Vivi mask grading as Miss Wednesday as well as Kuru, but they are not asked to join the crew as of yet. And now we have some pretty massive stuff to change though because after Whiskey Peak comes Little Garden, which is fine. This happens mostly as per we know it and Nami ends up catching her deadly disease. Now generally, this would be the obvious path to Drum Island. However, at this stage, we now already have a doctor being Crocus, a pretty phenomenal doctor as well, having once treated the pirate king directly. So this rather unfortunately means that Crocus is likely going to be more than capable of dealing with Nami's ailment, meaning that we bypass Drum Island altogether and move straight onwards to our abaster, which is kind of annoying because in addition to losing Chopper, we also lose Dr. Kuruha because in the series Luffy did ask her to join the crew as well. However, that's all pretty redundant with Crocus aboard. And I guess the biggest loss is of course the Straw Hat mascot, but to replace our cute reindeer, we now have a shrub in a box. So never fear, Cotton County lover Gaimon will be more than capable of carrying the series from here on out. So we reach alabaster much earlier than we would have otherwise, meaning that there is every chance that Vivi does arrive in time to meet with Koza and prevent the civil war on alabaster from kicking into full gear. Even if not though, the events are still going to play out with Crocodile being defeated, but in a grand twist, Vivi is not going to stay on the island. Instead, Boshi and Kuru are going to pursue their vague dreams of piracy and become regular members of the ever-growing Straw Hat Pirates. Oh, and let's not forget Robin. She will still stay away aboard the Going Merry and work her way into the Straw Hats as per usual. I mean, no, Luffy does not ask her, but he does agree. So that does not break our one rule. From here, we need to skip forward a fair bit because Jaya, Skypea and Long Ring Long Land offer us very little in terms of crew pickings. So we're going to end up on water seven to fix the Going Merry. And here we encounter an iceberg, a mere iceberg to be precise. And Luffy being Luffy once again, discovers a useful talent and then invites iceberg, or in Luffy's words, old man, to become the shipwright of the Straw Hats. And despite being both the mayor of water seven and the president of Galila, iceberg is going to have a rather abrupt change of heart and agreed to become a pirate. Hooray. Due to the tight timing, though the Cypherpole assassination attempt will still likely happen, leaving iceberg injured and thus unable to accompany his new captain to any soby in order to take Robin back. However, meeting Robin on the judicial island will be Luffy, Zorro, Nami, Saga King, Sanji, Gaimon, Ace, Crocus, Vivi and Karu. So this is probably still going to go fairly well or at least as well as it did originally, especially since Frankie is still very much present during these events. Speaking of though, Frankie is a bit of a question mark here. There is very little doubt in my mind that after any soby, he still would have built the Thousand Sunny for the Straw Hats, along with iceberg, they're now official shipwright. And I think an argument can be made either way. On the one hand, Luffy is a simple guy and he doesn't tend to double up on roles. Then again, Frankie did build the ship and the Frankie family is still going to beg Luffy to take Frankie with them due to his newfound global infamy. So with that in mind, I think what happens is that both Frankie and iceberg join the crew and become dual shipwrights, which is kind of nice because they were both apprentices of Tom, although it does leave water seven without a figurehead. No, well, that is not our problem. Sailing straight into Thriller Bark now we immediately encounter Brooke and he joins the crew. Very simple affair there. Although we do still need to help him out with the whole shadow business. So we are coming into conflict with Gekko Moria. Much more importantly than our penis necked wall, though, this means stumbling across both zombie tree and zombie unicorn, both of whom Luffy tries to capture and vigorously persuade to join the Straw Hats. Only this time, he succeeds. With both of the zombies all of a sudden having a change of heart, discovering hopes and dreams outside of their standard days of drinking peacefully in the dark, dank, foggy atmosphere of Thriller Bark. Really sad turn of events now though because as a result of Moria's inevitable defeat, zombie tree and zombie unicorn shadows are going to be returned to their original owners, meaning that these two members will be very, very short live Straw Hats, as well as I suppose the first case of a Straw Hat death that is not the going merry. And so at the end of Thriller Bark, the Straw Hats going to hold a respectful funeral for their lost crew members and are then forced to continue onwards, which takes us to Sarbadi where despite the fact that we have figures such as Ace and Crokers, the Straw Hats are still going to be defeated and separated by Bartholomew Kuma. But this is where things start to get really, really difficult and mostly because of Ace. He's been with us the entire time now. And as such, he has no association with Whitebeard and he might not even have a bounty of his own, although he probably does. Very importantly though, this means that Ace does not become a target of Blackbeard in any way, shape or form. And it actually kind of prevents the entirety of the Paramount War from even happening. For this event to occur, Blackbeard would need to capture another member of the Whitebeard Pirates, none of whom were as brash as Ace. So this whole scenario seems nigh on impossible to achieve, which also may potentially mean no time skip because Luffy isn't truly aware of just how weak he is anymore. And this also goes on to change a whole ton because Jinbei no longer needs to resign from the Warlords, nor does he actually meet Luffy and it's entirely possible that he never does. But because one piece is a story centered mostly around fate, I have to conclude that Luffy will decide on the two year training period. Otherwise the entire crew would be wiped out very, very early on in the new world. So if you do want that ending, then feel free to stop here, I guess. But if we proceed with the time skip and the crew reunite, all 13 of them now, we will then go on to Fishman Island. And quite frankly, I don't think Jinbei is there due to still being a Warlord. And even if he is, he has no association with Luffy. And it's very hard to see the captain asking Jinbei to join his crew without that background. But since that is massive, massive speculation, we'll say that the blood transfusion happens. Luffy asked Jinbei anyway, to which the whale shark has no excuses this time around. He just says, yes, Luffy-kun. We are far from done though, because Punk Hazard is now going to bring us a new series of members, one of which would be Kinamon's legs. And yes, I mean his legs specifically. Luffy did not, I repeat, did not ask the rest of Kinamon to join the crew. So we are just taking his legs. The rest of Kinamon can of course come aboard as a traveling companion, but only his legs are an official straw hat. On Punk Hazard, Luffy also asked random Centaur to join the crew as well. And this leopard Centaur dude is now very, very compelled to do so. So, okay, we now have this. So we then continue to make the alliance with law and proceed to Dressfrozer, where we do not recruit a new member, however, we do regain an old member. Having been sent there to investigate Dolphemingo's weapon trade, Sabo comes across Luffy and regains his memories, thus deciding to leave the revolutionary army and rejoin his captain. And in this universe, this would reunite what would be known as the OG monster trio, being Luffy, Ace and Sabo. And at this point in the story, that pretty much concludes the makeup of the alternate universe straw hats, going forward, this band would then have to tackle the rest of Dressfrozer, Zoho, Hawkeye, Kyland and now Wano, but there you have it. This is the crew that could have been if every character was as agreeable as Luffy wished they would be. Not quite the straw hats we know, but still an undeniable powerhouse team, nonetheless. But what do you guys think? Please do leave your thoughts in the comments below or even join my Discord server. And if you'd like to see more videos like this, then please do go and check out some of my other content or even subscribe to the channel for more glorious one piece business uploaded straight until YouTube feeds. But for now, this has been the Grand Line Review and I'll see you next time.


has given several different contributions to Schiff's theory. So today is going to talk about the Schiff's on T-topology. First of all, thank you very much for the invitation. And OK, I will talk about Schiff's on T-topology. I will start with a brief introduction, and then I will recall some basic notion that Bolivia also recalls in lecture about Schiff's on topological space. And then I will restrict Schiff's on T-topology. The idea is that in this setting, we can consider more objects. And we can study from really more objects and work with them. And after that, I will talk about a couple of applications on algebraic analysis. So I will see how, thanks to these new objects, we can construct enlarging a little bit of topology. We can deal with, we can construct functorially some analytical objects, which are very important. And then also, I will talk about some application to differential operation operators. So I will say that I will explain how enlarging a little bit the category of analytical object, we can distinguish a different kind of differential operators. It cannot be distinguished with a classical Schiff theory. OK, as I said, a Schiff theory is used in several mathematical contexts and has an application in various fields of mathematics. In the same context as semi-algebraic geometry, sub-anithic geometry, and a minimal geometry. And each of these theories has been very useful in various fields of mathematics, but each of these theories has some elements which was missing in the other ones. For example, in sub-analytic Schiff theory, we didn't have the notion of fibers or stalking fibers. In a minimal geometry, we didn't know about the operation or didn't know about how to construct acyclic objects. And the theory was to find something to generalize these various constructions. And so a media, which was introduced by Kashiwara and Shapira, but I think it is known by other mathematicians as well, was to consider less open and less coverings. So a very easy notion of site and extend the classical machinery of Schiff's to this context. And in this way, we can give a first, a fine description of the various constructions, so the sub-analytic, semi-algebraic, and the minimal one, using the notion of t-topology. So I recall briefly what is a Schiff. Again, I'm interested in Schiff's of vector spaces. So we consider a topological space in the field. A per-Schiff is a correspondence, is the data, corresponded from the open subset to the category of vector spaces, to each open associated vector spaces, and to each characteristic distribution, to each inclusion, we associate a morphism, which is the restriction morphism, which is compatible with the inclusions of open subsets. And that means that it is a contravariant function from the category of open subsets over topological space, where the arrows are the inclusions, to the category of vector spaces. So given this contravariant function, we can define Schiff, which is a per-Schiff, satisfying some gluing condition, which is represented by this exact sequence. That means that if we have a section of an open subset, which is zero on a covering, then this section is zero. And if we have a family of sections on a covering of an open subset, which glue on the intersection, then we can find a global section extending all of them. The classical example are continuous functions, because all we know that if a continuous function is zero on a covering, then it is zero, and then we have a gluing condition on coverings. So these correspond to define this basic example of a Schiff on a topological space. But there are objects which are interesting, but do not define Schiff's on a usual topology. And this example is this one, for example, bounded continuous functions. For example, on the real line, we can find coverings, infinite coverings, and bounded functions, which do to a function which is not bounded. For example, we can consider the covering of R, even by the open subset minus nn. Of course, each continuous function is bounded of minus nn, but it is not a bounded, not every continuous function is bounded on the real line. For example, the x to the x square. So its correspondence is not a Schiff. If we consider the usual topology, we have to change something. And on topological space, we have the notion of Stokes, which is represented by this inductive event, the inductive limit, the inductive limit on the section. The stoke on the point is given by an inductive limit or the section on the neighbors of the point. And the element of the stoke are equivalence classes. So an element of the stoke is represented by a section on a neighbor of the point. And two sections are equivalent at the level of stokes if they coincide on a neighbor of the point x. Stokes are important because thanks to them, we can check if two shifts are isomorphic. We can check if sequences are exact. And we do everything on Stokes. And the definition of a topological, a topological space, if we look at it, only depends on the notion of open subset and the notion of covariance. And we can slightly generalize this notion by choosing only a sub-family of open subset T and for each element of T, a sub-family of covariance satisfying some suitable hypothesis. And in this way, we can define a site, a very easy type, a very easy example of site because we just take less open subset and less covenants. Arrow size are always the inclusions, so we only have one arrow between two objects of T. And in this case, we can define pre-shifts as a contravariant factor from this new family T to the category of vector spaces. And we say that a pre-shift is a shift. If it satisfies the gluing condition, only on this restricted family of open subsets and covenants. So we still have the exact sequence of the gluing condition, but it is restricted to a smaller family of open subsets and a smaller family of covenants. Why we choose less open subsets and less covenants? Because we can consider more objects For example, we have seen the example of bounded continuous functions. And if we take topological space, we consider all the open subsets, but now we consider only finite covenants or better covenants, which admit the finite refinement in order to attain the site. And we consider the correspondence we have seen before from U to continuous bounded function of U on U. In this case, if we consider finite covenants, we can glue the bounded function to a continuous function, but we can also take the max of the bounds. So we obtain a bounded function with glue all the continuous function on the finite covenants. So if we consider less covenants, we have one more element which becomes a shift. In this case, a continuous bounded function becomes a shift. So with this idea in mind, we try to extend the notion of shifts or subanalytic or minimal semi-algebraic and other kind of sites. So we consider a topological space. And on this space, we consider a family of open subsets which is stable under union and intersections, such that a difference of two elements has a finite number of connected components and such that it is a basis for the topology of the topological space X. And starting from this, we defined the site XT where the elements are just the elements of this family of open subsets. And the covenants are the topological covenants admitting a finite refinement. So I wrote finite covenants in this sense. So it is important to have finite covenants. Examples of this family T are, for example, the family of semi-linear subsets in RN, so the open subsets which are defined by linear equations. The family of semi-algebraic subsets of RN, the subsets which are locally defined by algebraic inequalities. And the family of relatively compact subanalytic subsets of a real analytic manifold, which is the subanalytic site XSA. And this example I will use later for the applications. And more generally, if we have a minimal structure, the family of open definable subsets of this minimal structure. All these sets are stable under union intersection and have this nice property of having a finite number of connected components. So if we fix this family T of this property, there is a nice category inside the category of classical shifts, which is the category of coherent objects, which is the category of shifts, such that there is an aprimorphism from a finite sum of objects of T and such that for each morphism, if we find a sum of constant shifts on T, the kernel also satisfies this property one. So there are shifts which have a finite resolution on this constant object U. This is a full balance of the category of the category of shifts on X. And if we consider the natural functor of sites, which is obtained by the inclusion of the family T and the family of the open subset of X, we have a functor of direct and the inverse image from the category of classical shifts, topological shifts to the category of shifts on the site XT. We have direct and inverse images related to them. And the direct image is also a fully faithful inclusion, exact inclusion of the category of coherent shifts in the category of shifts on the site XT. So we can really see this category of coherent shifts as a subcategory of the category of shifts on XT. And we have an additional description of this category of shifts on XT because it is equivalent to the category of coherent shifts on T. And that means that an element of shifts, the category of shifts on XT, can be seen as an element of an ind object, as an ind coherent object. So a limit of a fantasy from a coherent to a set. And this category is also isomorphic to the category of left k-additive contravariant functions from the coherent shifts to the category of vector spaces. So a shift on XT can be seen as a contravariant function from coherent to mod k. Thanks to this, we also can construct in a very easy way the shifts on XT. So if we have a pre-shift, so if we have a correspondence from the formula contravariant function from the family T of open subsets to the category of vector spaces, it is easy to check if it is a shift or not. We assume that the input set is sent to zero. And we only have to check gluing conditions on couple of objects of this family T. So we have to check that this sequence of gluing conditions is exact only on couples of open subsets. If the second is exact, we can say that f is a shift on XT. For example, on bounding function, we just have to check that two bounding functions glue on a couple of open subsets, which is quite straightforward because we take the max of the two bounds. So in the case of shifts of T topology, we don't have the notion of thoughts because we are in a grotting topologies. So we cannot see if two, for example, if two shifts are isomorphic, only checking fibers on the points of the topological space X because even if this family T for my basis, so if this limit is actively the limit on the neighbor of the point X, the stocks on the points of the topological space are not enough to check if two shifts are isomorphic or not. Because, for example, if we consider continuous and bounded continuous function, we cannot distinguish continuous functions only checking or on stocks because on stocks any continuous function is also bounded. We consider the limit of neighbors on a point because any neighbor contains a component. So if one wants to consider only stocks on classical points, we lose information about the shift. We need more points. And these small points, we take these small points considering the spectral topology associated to this family T. So we consider the prime filters of elements of T. So filters such that the union, if the union of two objects is inside, one of the two is the filter. And if U is an element of T, we consider the U tilde of the filter such that U belongs to that. So in some sense are the filters containing, the alpha contained in U. And a neighbor of a prime filter is, this sense is U tilde with the U tilde element of T contained in alpha. And we call the X tilde the associated topological space. And in this space X tilde, the open subsets U tilde are quasi-compact. So they admit every covering, admit the finite sub-cover. I give an example because when X is equal to R and T is the family of finite units of open intervals, we can classify the points of R tilde. We have five kinds of points. The first one is the filter of U containing X. And this corresponds to the real points of the topological space somehow. Then we have the points which are represented more or less by left and right neighborhoods of the point X. So the point X plus and X minus. And we have to add the points at infinity. So if you want left and right neighborhoods of the infinites. So you see that we have much more points in the X tilde than in R, 30 line than in R. And thanks to these new points, we can distinguish effectively continuous function from continuous bounded function. Because for example, we can consider the continuous function one over X. Then it is not clearly, it is not bounded on each open subset of the kind of zero epsilon. That is a continuous function on zero epsilon. So it is not bounded on each neighbor of the point zero plus. So it is not X minus one. It does not belong to the stokes of bounded function on the point zero plus. But it is, it will also the stokes of continuous function on zero plus. So on the point zero plus continuous functions and bounded functions are different. But they coincide on each point of the kind one. So on each, if you want classical point of R. And if you want other example, for example, that function f of X equal to X, it is a continuous function, but it is not bounded at plus and minus infinity. Stokes at plus and minus infinity are different. But we have much more. One can prove much more. And sorry, X is a topological space, not the Riana 19 manifold. The category of shifts on the site XT and the category on topological space X tilde are equivalent. The category of shifts are the same. Level of topos, they are the same thing. So if we consider the category of shifts on XT and we want to work on stokes, we have to consider the topological space X tilde. And this is useful because sometimes to check something or to make some operation, we really want to consider fibers, stokes or fibers. And we have a good topological space making the job. Another important construction we can do on this site XT is the subcategory of T-flabby object. T-flabby objects are shifts such that for each couple UV, which U containing V, the restriction morphism is subjective. And this is very useful when we want to find the resolutions of funtors because this family is injective with respect to the functor. On F with F coherent. And this family includes all the section on opens. And with respect to direct images as well. Of course, once you have defined a morphism of topological spaces from X to Y and which is compatible with the family PX and TY on the topological space. So these T-flabby shifts are enough to derive and to find a resolution which are a C-click which are injected with respect to HOM and to the functor F-star associated to a continuous function respecting family's T to topological spaces. Okay, so thanks to this construction we can have a global approach generalizing subanalytics, semi-algebraic and and all these construction are useful in all these settings. But now I want to focus on the subanalytics side and talk to you about some constructions we can make on this setting. Thank you. Which have been part of my research. So we consider the subanalytics side as I told you before open subsets are relatively compact open subanalytics subsets and coverings are covering or relatively compact subanalytics open subsets admitting a finite sub cover. Okay, so we have the idea is that we can consider more object as the basic example of a bounded function in this setting we can consider for example, temper functions. So function which have a polynomial growth at the boundary of an open subsets subset infinity functions. Again, this is not a shift with the usual topology as in the case of bounded functions. So you see that when you want to consider condition at the boundary of open subsets functional spaces we depend from conditioning the boundary of an open subset they usually not define a shift if you consider the usual topology because we lose this condition at the boundary. But again, in this case I give you an example of this factor. For example, if we consider the real line we consider e to the x one over x and this function is a temper it is even bounded on each open subset un of the x between one of n and one with n natural but if you consider the interval zero one which is the union of this un the function e to the one over x is still well defined it's still a continuous function it's not tempered because it at around zero near zero it grows so fast so the growth is not polynomial anymore so it does not define a shift if we consider the r with the usual topology but it is possible to show that if we have two open subset which are subanalytic and we consider the intersection union and intersection we consider couple of open subanalytic subsets we have the exact sequence corresponding to the gluing condition so we have gluing condition on couples open subanalytic subsets and as we have seen in the general case if we check that a pre-shift is a shift on couples open subset of t in this case relatively compact open subanalytic subsets we obtain a shift so temper functional shifts are a shift on the subanalytic site and moreover in this setting we can also define the six grottendig operation we have all tensile product direct image and inverse image which are always well defined on a site but for a subanalytic shift we also have a proper direct image and one corridor and now let's talk about funtorial constructions so we see how using this funtorial operation we can construct analytical objects which are important we consider a real analytical manifold dimension and a family of submanifers and we suppose that they intersect transversely as a basic example we consider c2 and zi equal to 0 i equal to 1 or 2 and we need to construct the notion of normal deformation when we have an analytical manifold we can construct an analytical manifold x tilde m1 and an application from this x tilde m1 to from x times r satisfying its property d minus 1 over x minus s1 m1 is isomorphic to x minus s1 m1 times r minus 0 d minus 1 of c is isomorphic to x for its different from 0 and d minus 1 of 0 is isomorphic to the normal bundle of m1 we consider the restriction now of m1 to d1 different from 0 and we define m tilde 2 as the inverse image of m1 and we perform again the normal deformation of x tilde m1 with respect to this m tilde 2 we obtain another map and we define recursively the multi-normal deformation along the family i of submanifers as the x tilde m which is one with respect to m tilde we consider the 0 section of this deformation the inclusion s little s the set omega given by t1 to tl greater than 0 and we get this commutative diagram we have the inclusion of the 0 section and the set omega and we have the projection from x tilde 2x starting from this diagram we can construct the functor of multi-specialization which send a shift to an element of the 0 section of the normal deformation we take a shift we take an image by p we cut on the sets of parameters greater than 0 and then we restrict to the 0 section and I show you an example on dimension 2 to understand what is this specialization which corresponds to the setting considered by Majima when he studied asymptotic functions x tilde m1 is given by c2 with an extra variable and the projection is the multiplication of this new variable t1 on the first variable then we perform again the deformation with respect to m2 which is given by the coordinate zeta 1 0 t1 locally and we can construct x tilde adding an extra variable and the projection is the multiplication of the two variables t1 t2 z2 in this case the 0 section is equal to the product of the normal bundle over 0 on the normal bundles on m1 and m2 this is the construction we can perform on c2 and now we consider a shift of witness infinity functions which are functions which are bounded with the bounded derivatives on a relatively compact subanalytic open subset this is again a subanalytic shift on x we check a couple of subanalytic subsets of course we still maintain the bound on the function and on all the derivatives and then we consider the shift of witness holomorphic functions holomorphic functions which are bounded with bounded derivatives from these shifts we can construct funturally the shift of multi asymptotically holomorphic function multi asymptotically developable holomorphic functions applying to the shift of witness function the function of specialization specializing the shift of witness functions again when it is equal to c2 if we consider the witness holomorphic function c2 we can apply to this shift the functor of specialization and if we check on fibers it is the limit of the fibers of witness holomorphic function of the product of sectors containing a direction points of the product of this of the product of two normal bundles are direction when outside the origin and this shift this funtorial construction correspond to an analytical object which is very well known which are magemas asymptotics which are constructed starting from an asymptotic expansion which is given by this family of some sort of formal power series with holomorphic coefficients which is an approximate function and we say that a holomorphic function is only asymptotic redevelopable if and only if for any poly sector properly contained in a poly sector S we have this estimate the difference from S between F and the approximate fashion is founded by the power of Z1 and Z2 this is for each a sub sector containing this we have this classical analytical definition of asymptotic expansion and it is possible to check that a function is asymptotically redevelopable if and only if on each poly sector a each poly sector properly contained on a poly sector it is written so it is bounded with bounded derivatives and that means that the chief of asymptotically redevelopable functions can be constructed by specializing the chief of written holomorphic functions so if we know written holomorphic functions we can talk about asymptotic asymptotics on sectors and starting from written holomorphic function we can consider other subanalytic shifts which are the shifts of a flat written holomorphic functions and written holomorphic functions on Z is the the union of of the submanifers and we can prove functorially that we have this exact sequence of specializations and this is a this corresponds to the boring late exact sequences and we can probably we can prove it this sequence functorially by studying the of flat functions and written functions this is the exact sequence because when we consider the case of asymptotics we can check that asymptotics are the specialization of written functions the flat asymptotics are the specialization of written function flats on the axis and the third term corresponds to the consistent family of coefficients of magima and the fact that the sequence is exact corresponds to the classical boring rate theorem which says that if we have a family of coefficients we can find the function which is asymptotic to the family so we can obtain a general boring rate exact sequence for multi-syntotic developable function just for fixing the notation the family of consistent family is is given by in situ the family of F1, F2 and F12 such that F1 and F2 are function on the first and the second variable which are asymptotic to this F12 which are just coefficients and as a last application of some analytic shifts we can want to see how adding new objects on the category on adding new analytical objects we can distinguish differential operators so we consider the shift of things of differential operators locally a section is differential operator with holomorphic coefficients and holomorphic functions are a shift of the model so this is d x on holomorphic functions by derivation and multiplication of coefficients and we can define given the model on the ring of differential operator the complex which is the complex we obtain applying p to gamma and this complex have a homology and degree 0 and 1 and the degree 0 corresponds to the homogeneous solution of the differential operator and each one is the co-kernel so we have the solution of ps equal to g if you want and we see that two operators are equivalents if kernels and co-kernels are isomorphic so that means sol fp1 and sol fp2 are quasi isomorphic an easy example if we consider two operators zdz minus alpha and zdz minus alpha plus 1 if alpha is different from one we can easily verify that the solution of these two shifts are isomorphic because we can the solution of the first one is z to the power of alpha and to the second is z to the power of alpha plus 1 we won't go on to the other side or deriving and then dividing by alpha plus 1 so these two complexes are isomorphic and the operators are equivalent but for example if we consider these two operators zdz plus 1 z squared z plus 1 they have z minus 1 and x z minus 1 as a homogeneous solution so if we consider meromorphic functions and enable of 0 then the h0 of the solution of zz plus 1 is c times z minus 1 but for the second operation the h0 view is 0 because the exponential is not meromorph these two operators are not these two operators are not equivalent even if in this case the holomorphic solution are because x of z minus 1 is holomorphic in general if we define the definition of a regular operator a regular depending on the order of the terms and the derivatives of this fundamental theorem saying that if p and q are two operators which are regular as 0 in the neighbor of the region they are equivalent if and only if the holomorphic solution are quasi isomorphic so holomorphic solutions are enough to show if two differential operators are equivalent or not if they are regular so these of course have a huge generalization in the Riemann-Illberg correspondence by Kashiwara and Nebuchut what about if an operator is not regular in this case subanalytic shields can be helpful we consider the subanalytic shift of temporal holomorphic functions I said before that temporal holomorphic is infinity functions are a subanalytic shift now we consider holomorphic functions and they have a structure of a you have to add this lower restrict in order to allow multiplication with the coefficient it does not create any problem and we consider two irregular operators so they do not satisfy the if we check with the definition I gave before they do not satisfy the definition we consider z squared z plus one and z cubed z plus two the solution are the exponential x prime of z minus one and x to the x minus two and this is not really a theorem it is easy to check that there are open subanalytic sets containing the origin of the boundary such that hx a x of z to the minus one is a temper x of z to the minus two is not temper for example in the left half plane so in particular that means that the h zero of the temper solutions of z squared plus one and z cubed z plus two are different because the first one is given by x of z to the minus one times a constant and the other is zero because on this open subset it exists z cubed z plus two is not temper anymore so we can distinguish these two irregular differential operators thanks to temper the holomorphic functions and this is not possible with the holomorphic the two exponentials are well defined in the set where in a set not containing the origin and this can be generalized because there are some particular differential ordinary differential equation of degree one which can play important role in classification of differential equations which are the one which have as a solution the exponential of p z where z where p is a polynomial z minus one z to the power of one c of z different from zero all these operators are irregular and for all these operators the holomorphic solutions are not enough to distinguish them so we cannot distinguish them using the sheaf of holomorphic functions but this is a theorem of morando if these two polynomials are not multiple by a constant so for large family of polynomials the holomorphic functions are enough to distinguish the operators so we can find open subsets such that the solution on one exponential is tempered and the other not so thanks to temper the holomorphic solution so going to the subanalytic site we can find an object which now can be represented as a sheaf which are tempered functions and thanks to this object we can distinguish irregular differential operators we cannot distinguish with the holomorphic solution but we can distinguish a very big family of irregular differential operators and I want to add that this problem was completely solved with a very huge technique which is why Daniel first have to add one variables and then go to the category of enhanced sheaves this is a very technical process but you can solve completely the problem of distinguishing irregular differential operators and irregular differential operators and now just final slides about because I told about these topologies generalize a minimal sheaf theory as well and when we consider minimal sites we can perform all the construction but also we can develop the formalism of six grottendick operation thanks to the fact that we can work on fiber and for this a for various kind of minimal structure so we can define the six grottendick operations and we can define the six grottendick operation and thanks to the grottendick operation we can develop a very important formulas and computations so as I derive a projection formula the derived batch change formula coonet formula and develop the computation of these homologies and in this case it's different from the case of subanalytic shears where we could really work on the site and then compare with the classical sheaf theory where a lot of this formula were state in this case we work very work on fibers we really work on the associated topological space we work on fibers and the fact that topological space are good because they are normal we can perform on these constructions and ok thank you very much for your attention your attention thank you so much


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