Sheikh Muhammad Nasir, his biography, Muhammad Nasir D. Al-Mani, may Allah have mercy upon him, was that he used to go to the Haram? The Sheikh? Makkah. And anyone who's been to the Haram will know that people don't care, they'll step over you. To the extent, some of them are women. What are you going to do to a woman? It's a Moshkilla. So the Sheikh did not care who it was. Sheikh Muhammad Nasir, a dealer that did not care. He would go pray in the Haram and I believe, insha'Allah, we're going to study that more in detail. The Haram has a different ruling. The sutr does not apply in the Haram. It does not apply in the Haram. Why? Because it's impossible. But the Sheikh believed it's mandatory. That's why he said the general ruling is here. He used the general ruling. He said, The Prophet's speech is general, otherwise you didn't need to take it out because what he's point of view is. So the Sheikh, may Allah have mercy on him, would go to the Haram and he said in his biography when he spoke about it, he said, my whole Salah would be a fight. All the other fight. People won't listen. Ha ta'akhia, insh, sujood, people will step over you. Sujood, so it's not a matter of way of crossing by. So The man to the Sheikh was very, very solid. That's how he does his love of that. Sunnah, may Allah have mercy upon the Sheikh. So now we learn that it's mandatory to have a sutra. Brothers, these are matters you're looking at, see what was mine. Wa Allah, his sutra is very important. This is why Allah gives us honour and izah, when the ummah follow everything that they told to do. You see? So have a sutra when you pray. Also the Sheikh says after that, Wa yajibu, it is mandatory, ayyad nuwa minha, that he gets close to it. The only Nabi Salah al-Islam al-Bida'alik, according to the Prophet, what? Issue. So he gets close to it, according to what? According to where he can do his sujood. So when he does his sujood, the pulpit should be, sorry, the sutra, the sutra, the object that is there, his head is close to it. That's all it should be. The Prophet, and the Prophet sallallahu alaihi wa sallam, his sutra and himself, what was in between them both? Mu'marr-i-shat, where a shat, a goat, can go through. That's all it was, alaihi sallam. Very close to his sutra. How can you defend a sutra where it's over there? How?  So if the person has to be as close so he can grab, so it should be, to be honest, a reach. The person shouldn't be able to go around your hand when you do this. He should be able to go around your hand when you do this. He should be able to go around your hand. He should be able to go around your hand. So get as close as you can to your sutra. So you can defend it of course. And that's as I said, between the Prophet sallallahu alaihi wa sallam and the pulpit, it used to be that shat, alaihi sallam. Anyone who does that, he has come with the closeness that is required from him. The length of the sutra. The length of how high it used to be. The sutra should be high. So it shouldn't be, hey what? A book like this. On the ground. A flat book like this. Or an imam that you or a shemaag that you took off or a jacket that you took off. You just slide it down like that. The sutra has to be something that is vertical, something that is something that stands. Is that horizontal? This way it's horizontal in it. It has to be vertical. It can't be something that's horizontal. It is mandatory. The sutra is vertical. It has to be high. And on the ground. One hand span. One hand span. Oh Shibraini, all two. According to the Prophet sallallahu alaihi wa sallam According to the Prophet sallallahu alaihi wa sallam If one of you prays, places before himself Aimukherati Aimukherati is what? It is the thing that when the person is on his riding beast he leans on. That little thing. It's a little saddle naam. Mashallah and it's got a little thing, wooden thing that comes out of it. That thing reaches just above your the where your spine which starts. It starts from your bum, ha? So just in the middle of your back basically where your kidneys will be located at. From where you would sit. That's how high it would basically be. Shibraini naam. The Prophet used to have that. Pray to it. And do not have any consideration of whoever who goes around it. If the people are going around it, good. Mashallah. If they go inside it it's a mashkela. What do you do? Fight with him. If he tries to. The person will be directed towards the sutra directly. Don't put your sutra to the side. Are you with me? Don't put it. Your sutra and you are so the qiblah is there and the sutra is in between you and the qiblah. It's in the direction towards the qiblah.  Because that is the apparent order that is required towards the sutra. But to place the sutra on the right on the side. To move it to the right of the side. Left or right. That is not authentically. That is not right. So that's something that many people may do at this current time that we have. It is permissible for Salah to a stick It is permissible for a stick that a person sticks on the ground and he puts it on the ground or other than it. Or a tree. So what's he trying to say? Something that you stuck to the ground. That's what you done it.  You put it in and you play towards it. That is permissible. A stick is not going to be that thick. So the thickness of it it really doesn't matter. It's the height that matters. The thinness of it doesn't really matter. Or a tree. A tree is big. Or a tree. A stick and the tree is going to be what? The tree is originally from the ground in it. So it's rooted from the ground. So anything that just came out of the ground and anything you put into the ground no problem, it's the same. The thickness and the thinness is also not a problem. A wistiwana tistiwana means a pillar.  A person is lying down. They are one sheep. She is a sultana. And the prophet used Aisha. So Naam used Aisha to grab her leg every time he was doing sujud alasari salam. Using your wife as a sultana. Not a problem. And also a riding beast. Your beast. You use him. Even if it's what? Naam, even if it's a camel. Why would the sheikh say even if it's a camel? Are you allowed to pray in a place of the camel? It's one of the places prohibited to pray in the toilet. The house, the place where the camel stays. So the sheikh is saying that some people may think that if the camel is not in that place he's outside on the tree. He may not be permissible to pray using the camel as a sultana. It's just a place that's prohibited, not the camel. Can you only use your wife as a sultana when she's on the bed? Naam, if she's on the floor, she's lying down. You use her Naam. You can use her back as a sultana.  Can you stay here and I'll iterate towards you. Towards the sultana. Use the sultana. Don't run after the sultana. Don't run after the sultana. Naam, it's true because some people they do the sultana going. We mentioned that. The prohibition of the Salat towards a grave wa laa taajoozu it is not permissible. As-salah to pray a Salat. Ilaal qabori towards the direction of a grave. Moutlaqan whatsoever. Sawa'an kaanat qabooran li'l anbiyaa wa gheireem. Whether that qabr is a prophet or other than them. Doesn't matter how holy this person is. And the prophet sallallahu alayhi wa sallam has made so much effort. Five days before he died you can imagine a person in the last moments of his life he talks about the most dangerous things. If Allah mentions Surat al-Nuh wa qaaloo laa taqidun Surat al-Nuh Allah says wa qaaloo laa taazarunna aaliha taakun wa laa taazarunna waddan wa laa sawa'an wa laa yawuth wa ya'uqa wa nasra They were five righteous men and they were what? They were worshipped. So it's these tamatheem and these people that have been on it highly too much taking it out of proportion. It's one of the masyakins. So it's not permissible to pray towards the sorry the Qubur at all whether it's a prophet's grave or other than that. Now the shik goes into it. He says the prohibition of crossing a person who is praying even if it's in the haram. Even if it's inside the haram. He mentions that. It is not permissible.  There is no difference. There is no difference between the harami mosque in Makkah and other than those in that mosque. Does it matter whether it's Makkah, Medina or anywhere it doesn't matter? So all of them All of them are under the All of them are what? That is under the prohibition. Any masjid, they all fall under the prohibition.  But even if you take this minimum which is what? 40 days. 40 days, or 40 Salahs, or 40 years, it is something very not easy. So it's big that the person does this. Naam.  You see? Even if it's in the haram. It is not permissible for a person who is praying. It is not permissible for a person who is praying to a sutra that he allows any person to go in front of him. Why? Because of the hadith that I just went by.  Ha, the hadith.  And try to push him as much as he's able to. And another narrate, what did the Prophet said? The Prophet said twice. Stop him, stop him, do not let him. If he refused, let him fight him. But when he's shaytan, that person is shaytan himself. Naam. Push you in front of me, walking forward in the Salah, to walk forward. To not let anyone cross you.  They're still coming again. It doesn't get it. So what do you do now? Or the person is coming with a full speed, you can see they're running.  It happened to the Prophet, when the sheep came running fast. And the Prophet saw that the Prophet was coming. He saw it coming. What do you do? You run to your sultana. So he goes behind you. The Prophet did the Alayhissallaah to Salam. It is permissible. The person to take steps, one or two steps in his Salah. Because this person that is running to you is not burdened. They're not sitting for crossing you. It's like an animal. It's not burdened. He's not burdened. He's not burdened. He's not going to be held accountable. He doesn't fall under the 40 that we mentioned. But you still have to take the responsibility of stopping him. That is your job. Even the child. Naam. The person has to run to his sultana. But if the child is playing on you, it's a different ruling. Nah. If he's playing on you, it's a different ruling. Nah. If he's playing on you, it's a different ruling. Nah. If he's playing on you, it's a different ruling.  If he's playing like the Prophet As-Salaamah did. Like in his crossing you by like that. The Prophet As-Salaamah would refuse him.   You can pick the child up. You can put him in front of you. You can carry a child in the Salah. Ma'aqta us-salata. Anything that's going to cut. Anything that's going to cut. The Salah. Anything that's going to cut. The Salah. Ma'aqta us-salata. Anything that's going to disconnect the Salah. Wa-inna min ahmiyyah. Wa-inna min ahmiyyah to sutrati fis-salati. The importance of the sultra for the Salah. And not the hulu that it barricades. Or blocks. Baina al-musalli, the prayer. The one that's praying.  Wa-baina al-musalli, the destruction of the Salah. One of the things that the sultra does. Is that it blocks you. The prayer. The one that's praying. And anyone trying to tamper with your Salah. The sultra helps you from it. Some people who harm your Salah. And he will mention what it means. Bil-muruni baina yaday.  Bi-khinaafil lavi lam yatakhidha. Opposite to the one that doesn't even take it. The one that doesn't take the sultra. These things will nullify his Salah. Nullify his Salah. Remember. There are those who can nullify your Salah. There's no sultra.  The Salah just go. These ones, if you have a sultra. And they go around the sultra. The sultra has protected you. From anyone to be able to harm your Salah. Fa-inna yaqta'u salataw idha marrat baina yaday.  Al-bariga. That has reached age of puberty. If she goes in front of you. The Salah will be disconnected. Wa-kadaa li-kal-himaraan usadaa. Donkey. Wa-l-kal-bul-aswad. And also the black. The black dog. Naam. And also the. Black dog. Those three, when they come into the Salah. What will they do to you? Your Salah does not. They go like him. She has to be what? Al-bariga. Haith. She reached age of puberty. A little girl. No, no problem. If she goes in front of the Sotra. Is when she nullifies your Salah. But if she goes around the Sotra. She can. Around your Sotra. No problem.  In front of your Sotra. Your Salah is void. It doesn't exist. That Salah is not accepted. The same is the donkey. And the same is the. I shall refuse this hadith. Now. Men. Women. Donkey. And black. Dog. She refused it. But she didn't refuse it. Because the Prophet SAW. Did not. She wasn't mockering that. The fact that the Prophet SAW. Said this and she. Like. How could he put us like this? That's not she was refusing. The authenticity of the Prophet. Having to have said this. She rejected it. But for us. Has the authenticity reached us? We take it. Can't question it.   And we are to intention. So we've mentioned. And the standing. And we also have mentioned. The Sotra. We've mentioned the Salah with shoes. Praying on the. Plane. And. Ship. Combining between standing and sitting. And. And other things like that. Now we're going to go on to. The third chapter. And we are to intention. Wala buddha. And it's necessary. Lil musalli. The one that is praying. Min an yanwi. An is salati. Let he come out of her. That he intends. For the prayer in which he stands for. Wata. You know how. And then he specifies it. Be called. He in his heart. Kaffar. The door. I will ask. Such as the mandatory prayer. Zuhur. And asir. O sunnati hima matala. Or if it's the sunnah of any one of them. Wa hua sharatun. It is a condition. O ruknun. Or it's a pillar the intention. Wa amma talafud. As for the utterance of the intention. With your time. Biha bilisani. Fabidah. It's an innovation. Mughalifah. That opposes. This sunnati it opposes the sunnah. Wa lam yakun biha a'hadun. No one has ever said it. Min matbu'ayil muqallidina min al-a'imah. From the rightly guided. Fokhulafa al-rashidil. Fokhulafa. Sorry. From the. Fokh al-a'imah. Abu Hanifah. Nimam malik. Nimam shafi'i. Nimam Ahmad. And other than them. No one has ever said it. The intention to be pronounced. Loudly. None of them have ever said it. None of them have ever said it. So here the sheikh is talking about the importance of the intention. The intention has to be said in the heart. The person has to distinguish between the salas. So remember, if it's a dhoor and it's a asr, you have to distinguish between the two. If it's a mandatory prayer and it's a highly recommended prayer, then you have to distinguish between the ruling as well. All of this takes place in where? In your heart. The sheikh did mention whether it's a condition or it's a pila. Both of them are present. At takbir. Takbir of the salah. To say Allahu Akbar is at takbir. After he intends the salah. So the sheikh is walking us through the house. So you face towards the qiblah. You face towards the qiblah. You've stood up.  You came with the intention in your heart. Now you're going to open the salah. So he's walking you through it until you finish the salah. Each point. Like that. And near to we finished it. At takbir. The saying of Allahu Akbar. Then after that. Yes tafti hus-salata. The individual opens the salah. By qawlihi by saying Allahu Akbar. Wa huwa ruklun. It's a pila. By qawlihi sallallahu alaihi salam. And it's a pila. And the evidence for that is the speech of the messenger. The key to the salah is purification. And for it to become haram from you to do any actions. Or anything. It becomes haram from you to do anything you just want to do. By speech and action. By what? At takbiru by saying Allahu Akbar. Speech becomes haram from you. Having a conversation with your friends. Going to wherever you want to. Just walking around. All of these become impermissible from you. Actions becoming impermissible from you. And for those actions to be permitted for you. Is at taslimu by leaving the salah. Leaving the salah. By saying as-salamu alaikum. And as-salamu alaikum.  So the prophet said. The key to the salah is purification. The prohibition is by saying Allahu Akbar. The permissibility now. It's after saying as-salamu alaikum. As-salamu alaikum. Wa la yarfa'u s-sautahu. The person will not raise their voice. By takbiri. With the takbir. In all the salah in every prayer. Unless he is an Imam. A man comes into the salah. Right at the back. And he says Allahu Akbar. And the people think the Imam said it. And they just go to record. People have seen that happen. Why would you even raise your voice? Who told you to raise your voice? Say Allahu Akbar. You don't raise your voice. Who is the one that raises his voice when he says. The Imam. The Imam is the one that says. Allahu Akbar. And everyone. As for you. No. Wa la yarfa'u s-sautahu. The person doesn't raise their voice. With the Allahu Akbar. In every raka'a. No. Except if he is what? An Imam. Except if he is a Imam. So the person says. Allahu Akbar. Say it. How much he needs. No. Wa yajouzu tablighu al-mu'addin al-takbir al-imami ilan nasi. Wa yajouzu it is permissible. Tablighu the conveying of the mu'addin takbir al-imami. To convey the takbir of the Imam to the people. Say the Imam because Allahu Akbar. This person is also. The mu'addin is allowed to say. Allahu Akbar. To convey it to the people so that it can reach them. Because there weren't microphones. There wasn't that microphone. So he is allowed. Can everyone just do it? What did she do? She just become a folder. Everyone just saying that once. The mu'addin is the one that does it. So the Imam says Allahu Akbar. The mu'addin says Allahu Akbar. إذا وجد المقطظة لذلك. That is if the need is there. كما رضي الإمام. Or the Imam is ill. Even if he used the microphone. He is ill. He is ill. His voice is not clear. وضع في صوتين. His voice is very weak. Other people that are playing behind him are too much in number. So it's not just a fact that the microphone is not there. Sometimes without the microphone the Imam can't be heard. So it shouldn't be shouted out. If it passes on and it dies out on a particular point. The ones that can just fade out they carry on as well. نعم. The haram is in number. The haram is that. It's the number. The first sound is not very hard. It's clear as the second one. And sometimes people think did I hear that? Did I hear that? Did he just say it? Was that the Imam that just said Allahu Akbar? Some people say that to themselves. But then the mu'addin says it. It's clear. It becomes clear that it was the Imam.


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I think what's the stat, one in every five companies has SAP, something like that. I'm John Furrier, the founder of SiliconANGLE.com, that's the reference point for tech innovation and I'm joined with my co-host. I'm Dave Vellante of Wikibon.org and we're here with Fawad Zakaria and we're talking mobility, a theme, John, of the show. I mean, we've been talking on SiliconANGLE.com about cloud mobile and social for a few years now. Mobile, it's the obvious, home run, everyone's on mobile, gaming's moving to mobile. Mobile is the preferred platform. So talk about the trends that you're seeing here around some of the developments and some of the business changes around mobility. Thanks John and Dave, thanks for having me. Mobility of course is an exciting area for us. So lots and lots of activity here at Sapphire around mobility. We've got our platforms, we've got folks developing applications on them. As most of you know, mobility is just everywhere. Consumer mobility is really what's driving a lot of our activity but lots of what's going on in the consumer world is being adopted by the enterprise and so the mobility products increasingly are in demand in the enterprise. People are coming and asking the IT departments for mobile stuff and we're really helping transform the way businesses function by moving a lot of applications that people were using on laptops, other kinds of devices on newer and fancier mobile devices with user experiences that are really transforming user experience. So you're seeing lots and lots of stuff in Sapphire, lots of customer interest, lots of customers really doing both security which is really changing dramatically but also all kinds of new use cases that mobility is enabling for the first time. The folks out there that are watching, everything's on Facebook this week with the IPO coming, huge valuation, more billionaires being minted around the world, mostly in Palo Alto where I live in, which is good for real estate prices that schools will be overcrowded soon when they start having Facebook children as we say. But the point is that that's the social network in the consumer space and with Apple with mobility with the iPhone, the App Store concept, iTunes, everyone understands that and developers have made huge strides and dollars and made a lot of wealth, Instagram and today in a few. Your CEO, Jim Schnabe said that the next wave of innovation is going to be around business social networking, connecting people with people, people with businesses with businesses. You guys run an App Store, huge opportunity, can you talk about that opportunity and why it's such an innovation priority for you? John, not surprisingly, I couldn't agree more with Jim on this. Obviously, mobile is really enabling a whole new set of connections and social is a part of that revolution as well. So, App Store for us is a dramatic way to get the community involved to really help them gain access to over 200,000 of SAP's customers. Everybody is familiar now with the store concept, everybody goes to the various favorite stores, download their apps, their Facebooks, their games, all the various distractions of modern life, but increasingly it's being used for productivity in the businesses and so at SAP we have created a really cool experience for the enterprises around the store where we are getting all the entrepreneurs who want to innovate for the enterprise, for any application in the enterprise to gain access to that enterprise customer and in that instance the app is a wonderful channel where all our customers will go, will drive traffic to those and so the best ideas will get adopted by our customers in a totally new environment which of course is very unusual for SAP customers to realize but the App Store creates a completely new environment for us. So, Fawad, what are the metrics you're looking for that are success maybe number of apps, obviously, right? Absolutely. That's a big one, but talk about some of those shares. So number of apps is certainly one of them, but of course the number of apps itself doesn't mean much if those apps are not being engaged. So what we're really looking forward to is how many downloads do those apps get, which of these are most popular, which are the ones highly rated by the community, how much does the community engage in building newer stuff but creating value on top of stuff. So I would say that ultimately our measure is going to be the amount of money our partners make and SAP makes from those particular applications but that's going to be determined by how many apps there are, what kind of value they're creating, how many people are using them. So you've heard us articulate a billion users, that's really what we want to get to. We know that mobile is the path to a billion users and mobile, if we get to the billion users, is going to make our partner ecosystem tremendous amounts of money and ask a lot of money.  I guess you guys are going big. You know, one of the metrics that I love that we hear whenever we do VMworld is they say for every dollar spent on a VMworld license, $15 is spent within the ecosystem. You don't hear that a lot from other suppliers but you're not using that specific metric but you alluded to something similar just then. Is that something we're going to hear more of? You're going to hear a lot of that from us. In our traditional environments, people heard the number one to eight or one to seven or one to ten all the time where the service providers were making that amount of revenue for every dollar of license we were generating. They're very confined though. Very confined and frankly that's the model that we're changing. We're fundamentally altering that model because customers like fast time to value, they want instant gratification. That's what mobile is. That's what all the newer applications are like. And in this instance, our success is going to be measured by how many of our partners build businesses on top of our platform. So the more of them that put applications that we can make successful into our channel, not just today's SAP customers but tomorrow's SAP customers. You're going to hear lots and lots from us about that and we're going to do whatever it takes to make that ecosystem successful. You guys are actually a great technical DNA. We're going to have our founder on. We're going to get the legal system juiced up a little bit. Right? It was born on this day when a million were developers. We had to create incentives, right? So there's a huge opportunity in the enterprise. We had it so early. Now, what are you guys doing to get the word out? How do you attract those developers? How do you educate developers out there that we're talking to right now?  We've got its money on the enterprise. How do you convey that? So John, great point and I would make two points to it. So there is not only money to be made here but we're also to those developers who might be listening out there making it easy for developers to develop the way they want to develop. As long as they develop cool applications we just announced three partnerships with Centra, Accelerator, and Adobe PhoneGap which allows developers to write on any platform they choose to develop against the SAP platform and make money through SAP customers but develop the way they want to develop it. Developers care about two things. Distribution and monetization. What do you offer them? What assurances can you share with folks out there that you will take care of them? You don't need to talk charity but if they develop a good app they want to have comfort with distribution and cash. And we really want to make them as much money as they can. We only take a small referral but we're going to encourage. We're going to go market. We're going to send them off to as many SAP customers as we can and that's where the app store comes in. So that's going to be our primary channel. You're going to see us promoting that a lot and you're going to see partner apps that have demand and that customers want to rise to the top. And where can they go real quick in the URL or websites or interesting instances? So right now on the top of my head I don't have one but we just launched the Jumpstart program this Sunday and we have a landing page for developers where they can go and learn all about our mobile as well as our store and how the process works for them to develop an application and put them up in a store. So just search the web. We had the venture guys on earlier. We used to have heard about the industry.  Absolutely. Both great news for developers and for entrepreneurs if you have the success of LinkedIn I say I'm bullish on the industry. So we'll plug here. The success of LinkedIn to me is that it's a free text to what I see on the left-hand side of a tsunami of innovation and money making is going to come into the business side and expect their app store and price to price. Absolutely. You've got one iTunes, right? You've got a, you've got a, it looks like you've got a place to place all their iTunes. That's exactly. And there's store.sap.com I believe is that that's you guys right there. But you need a, you need a Silverlight download and I didn't do it. No, no, no, this is now an HTML5 so there's a new app already out. Yeah, okay. Which is a, which is a fantastic user experience. Coming soon. Yeah, I'm excited to see the HTML5. Thank you very much. It's wonderful. We've got this TIL coming on and Oliver Busman coming on. Thank you very much. Thank you very much. Thanks Sean. Thank you this guest. I hope it's great.


Awesome. Thank you, and thanks for putting this on. I believe the world will be a better place. I should also give you a renormalization instruction for any criticism that I might utter.  And I think it's not interesting.  And not trust it.  So I'm actually not going to talk about analysis pretty much at all. I'm going to talk to you about why we need to worry about analysis. So I'm also going to try and sort of ground us in a big picture of what are we actually doing here? So I'm very happy that somehow the term I think that I coined regularization has taken root but I want to tell you what that actually means from my point of view. So when you regularize a modular space, well, and when I say modular space, I'm usually going to mean pseudo amorphic curve modular space. So if you're given a compact modular space, which will always be called M bar, usually of J curves, I would like to assign to it in some sort of unique way, well, one or two things. So in the Grom of Wittenwald, I would like to actually give it a fundamental class. So, right, but then of course the question is what do I mean by a fundamental class in the homology of something that might just be a compact metric space? Right, so if this was a manifold, it would have a fundamental class. If it is not a manifold, we don't know what its fundamental class is. But the nice thing with trying to actually construct that right in here is that from the modular space, we have evaluation maps in Grom of Witten. Right, and so if we had a class in there, we could push it forward directly with evaluation maps. That's why I think that's the cleanest way of saying what we want is to make a homology class straight in there. So, this is really the Grom of Witten case and pretty much in all other theories, Fleur, SFT, anything infinity related, you're going to expect sort of your modular space to have boundary and so really what you want to do is you want to somehow count things and then get a relation from the boundary and so what we'd like there is some kind of cubordism class of closed No, not closed but compact things. Right, so here you could get this from maybe a compact manifold. Well, at least that would have a homology and how I get it into M-bar is not so clear, but here I'm right. I'm not going to say closed because they're going to have boundary compact manifolds and right, in general, they might be weighted branched and they're going to come from some choices. I knew that Chris Wendell up here had up here, but then unfortunately somebody diligently removed this, so your task for this next hour is also to figure out what's completely different in the philosophy that I project here than it is on Chris's board about abstract perturbations and I'm happy to discuss that in office hours. So right, so really these manifolds are going to come from some kind of choices of perturbations, new that we'll talk a lot more about. The main thing is it's going to have boundary in general and usually also then corners and the algebraic identities usually come from identifying the boundary with some kind of fiber product of products of the modular space itself. So if this seems totally alien, maybe think about Fleur theory, and what I'm thinking about here is I'm packaging all Fleur trajectory spaces in this M-Barnieu. Right, so no matter between which critical points and no matter between of what index and then this just says that somehow the boundary is given by broken ones. Yes. Yes. Well indeed, yes. So but that's why this is the philosophical overview. But I could do this on a certain up to a certain energy, and then this would be true. So good. So this is my goal. This is what I mean by regularizing a modular space, assigning to it something that I can count or that I could integrate over and that has some kind of nice boundary structure that gives me the algebraic identities I would like. So I don't need to write this. What are my tools here? Right? So if I'm an algebraic geometer, then I describe J curves on my modular spaces are really modular spaces of sort of sub varieties or sub not manifolds, but they're sort of subsets of an ambient algebraic variety that are cut out by some algebraic equation. And so then I can do algebra to describe my modular space. And the thing that Gromov realized was that, well, if my J isn't integrable, I can't quite do algebra, but maybe I can do exactly the same things that algebraists do with PDE methods. So that's why there's a lot of analysis here because all the magic that you can do with algebra we now have to do with analysis. However, we always have somehow this intuition that everything that's true in algebraic geometry should be true in symplectic geometry. Because the word of Gromov written in this was later based on the paper by Gromov. Right. Okay. Yeah. So some some things flow back. Yes. Catherine, did you stay away from like the last four inches? Yes. This. Okay. Yeah. Awesome. Good. Yes, I can. So let's see. What is my intuition here? So, well, I should I should say something, right? So first of all, I'm going to do an analytic description of M bar, namely in terms of solutions of a PDE. But then there are two other things. And so I'm just going to roughly think about having a fixed Riemann surface and a fixed almost complex structure on a symplectic manifold. But I have to do two more things. I have to mod out reparameterizations because as an algebraic geometry, I'd like to actually count the curves, not the maps. Right. So an image in M bar, I could parameterize in different ways by maps. And so I want to quotient out by that ambiguity. And then we're always going to want something compact. So I'm adding Gromov compactifications. And so maybe just as a side note to the examples that Duzer gave, right, the beautiful examples of non compactness coming from Mark points running together. However, that's not the worst of our problems, right? So usually really the hard problems with Gromov compactification come from energy concentrating. And that's where you get the sphere bubbles that are then have an unstable domain.  And that's sort of really a PDE blow up situation. So you really do need this Gromov compactification even if we have no Mark points. So if that's how we described M bar, then we can do, well, two things. So first of all, as you've already seen, I think in both talks, there's a more or less local or global Fredholm description together with gluing maps, which we'll sort of start talking more about. And so very roughly what that does is it somehow, this is a lie, but the intuition is that you're writing your modular space as the zero set of some kind of section. Actually, mainly the section here is a lie. But let's pretend, right? So at least to some extent that seemed to be the description. And then the reason you believe that these things should be things that have a fundamental class, I believe is really this finite dimensional regularization theorem, which says, well, let's pretend for the moment that my nice compact modular space of J curves is actually cut out from a finite rank vector bundle over a finite dimensional base. And let's say that's even smooth, I guess. Let me make it an oriented vector bundle. Then, well, first of all, even without the section, these things have an Euler class, right? So that's just the Euler class of the bundle, but you can pull that back and let it sit in the base. And I guess the theorem here, well, you can construct that with algebraic topology, or you can construct it by taking a section and perturbing it. So there's various methods of constructing Euler classes even. And so I think the main one that we talk about in symplectic geometry is to do it by perturbations. So I would like to be able to just say that my Euler class is actually maybe the fundamental class of some perturbed section of the zero set. So that would then sit in that homology. I believe I should actually add, oh right, so usually you need the base to be compact to get an Euler class, at least. But I believe you can do this as long as the zero set is compact. What do you mean as a homology class? Right, yeah. Right, otherwise it's just not unique as a homology class, there's no chance you'll ever have that. And also, right, so I'd like B and E to both be finite dimensional, but the finite dimensionality is actually not the problem. I think you could also take them infinite dimensional and then you would just need the section to be Fred Holm. And then the same theorem is going to hold with exactly essentially the same proof. Right, but what do I mean by this really? So how do I construct? What do I mean by having perturbations that give me this Euler class? And this is where, right, this is where the word generic comes in. That's what I was going to strike in big and red from Chris's board. Because really, so people always say take a generic perturbation or a generic J. So if you give me a J, how do I know whether it's generic? Generic is not an adjective that goes with a single element of a set. Somehow we say generic when there is a commiga set and then the elements of that set are the generic elements somehow. And so yeah, so I think that's one danger because people often say generic in situations where it's not even clear what the Banach space should be from which you're taking something.  But I actually don't quite understand why we're so focused on having sort of a commiga set or a large set or full measure set. So what I'm going to say is really all you need is a set that's not empty. And in fact, sets of second category are non-empty. And in that sense, that's a perfectly fine thing to prove when everything goes through if you just prove that something is second category. I just don't know how you ever prove that something is second category without proving that it's commiga. But anyways, so there is a non-empty set in the smooth sections. So that for any perturbation in this set, so these are the good perturbations. The perturbed section is not just transverse, but its zero set is still compact. Yes, we'll do. Thank you. So that tells me that it gets a smooth structure and if that's compact it has a fundamental class and then I can embed that into B and so I get a homology or I get a cycle there. But now what if I take two different perturbations? I would like them to actually have the same fundamental class. And usually we prove that by proving that the two zero sets are co-bordant. So people often talk about the transversality problem in holomorphic curves. And transversality is really just this bit whereas somehow regularization for me, when I say regularization I mean the whole package. I mean transversality, preserving compactness and then having uniqueness up to co-bordance. So if all you care about is that the set P is non-anxie, how do you know there are two sort of disjoint sets? Right, yes. So I need that to be sort of canonical. Right, I mean I want to, so right usually actually it's not, so usually the statement is not there exists the P, right. Usually the statement is P which I define as something, so it's a specific set. And I don't need to know that that's commiga, I just need to know that that set is not empty. So for example in Chris's theorem I think the statement is going to be that it's a set of J's in this curly J. Right, and the J's are defined exactly by some transfer, by regularity, by subjectivity of the linearized operators. Can I make one comment, so it's not that I disagree, but when I define what conditions those J's have to satisfy, it's almost always going to be that it satisfies a countable list of conditions. Right. And so for each of those individual conditions in the list, it is pretty important to know that there's a commiga set, because I need to know that the intersection of those sets is not empty. At the end I only care that it's not empty, but how else would I possibly know that? Sometimes you have some kind of special, in fact if you want to prove like most people's, your homology kind of things, you have something that you know, and you want to deform that thing that you know to, and then you want to have some kind of complex, almost complex structure that's near a given possibly degenerate. Right, sure, sometimes you'd like to know that kind of density. This is just for existence of a possible definition, yes. Right, I mean you can also, again, you don't need this to be dense, you just need to have it be dense near a given point. Right, so you could take a set and say, I fix the support and I fix some sort of size of the perturbation, and no matter how small the epsilon and delta are, the set is always non-empty. Since which point, you don't know which point you're going to need in the future. Yes, but I mean, right, so the statement that given any J there is an arbitrarily close by J that is regular is also different from proving that something is commiga, right, so. Catherine, while you're explaining philosophy, why do you take a perturbative definition of the order of class as opposed to like product in the grass, money, or? Right, because I'm going to want to generalize this in such a way that it applies to holomorphic. I mean, that definition wasn't anything, right, it could be a topological space for that definition. Let's discuss that once we're sort of deeper down in the actual application. And actually, right, well, challenges are going to come momentarily and then we can talk about whether different approaches will solve that. What was I going to say, P? Right, so in this case here already, I think the set, so I think Assad's male theorem here would tell you that, you know, the set of ones that, so if I defined the set by the ones that are transverse to zero, I would indeed get a commiga subset in there. However, it's not clear to me that for all the perturbations for which I have transversality, I'm also going to preserve compactness, right? So, in fact, I think I could cook up a counterexample. And then it's also not clear that you're always going to have uniqueness up to cubotism, so it's often you have to sort of fiddle some extra things in here in order to get these two other items. And that's the magic of perturbing J, somehow, right? No matter how you perturb J, you're always going to have grumov compactness and somehow between any two Js, there's always sort of a nice family for which you can run another grumov modular space problem and get this cubotism. So, you already see that somehow perturbations of J are nicer even than perturbing finite rank bundles. So, right? So, the history of the subject really is that people understood, okay, you have these local Fretholm descriptions. We have this perturbative description. In fact, we have a perturbative description of the Euler class of an Orbi bundle. That's what a lot of the papers took a long time to explain. And then how hard can it be to patch this thing together? That's literally, how hard can it be? That's literally as much as there was written in literature. If there's any justice in the world, then this directly generalizes in such a way that it applies to holomorphic curves. So, and I have to say, I believe that. I really didn't think it would be that hard. However, there are a couple of challenges here. So, one is sort of more a comment on the Kuranishi approach. So, if you really sort of forget about the sort of big D bar operator, you just remember that you have local Fretholm descriptions, then you can write your modular space as kind of a union of zero sets. Well, let me drop any kind of isotropy here for the second. So, some kind of transition maps. And so, there's a whole lot of local Fretholm operators, who zero sets will be some part of my modular space. And once they're Fretholm sections, I can even take a finite-dimensional reduction. So, I could either make these Fretholm sections or if you don't know what a finite-dimensional reduction of a Fretholm section is, you should ask someone this afternoon. It means you can replace, if this is Fretholm, you can replace the whole thing for the purpose of the zero set by a finite-rank bundle over a finite-dimensional space manifold. And right, gluing maps also fall into sort of the finite-rank sections picture. Right, so you have local nice descriptions and you have some kind of overlaps, right? And you would think that you could make the Euler class locally and then somehow patch the Euler classes together, right? How hard can it possibly be? Or you can think a little bit more globally, and that's what both Chris and Duzer did, is you can actually look at the zero set of the Debar operator on some space of maps, right? So, here the base was maybe maps from sigma to M. There's some bundle over it and D bar is a section that cuts out most of my modular space except, right? Then I need to quotient by the isomorphisms of my base, of my domain, and then this is usually still not compact, so I should somehow take a compactification in some Gromov topology. So, again, I'm seeing the zero set off a Fredholm section here, and with this sort of Sartre-Smale theorem, you know, I can perturb, I can make this a manifold, but as someone asked in Chris's talk, that does not mean that this isomorphism group still acts on it, right? If I put in some kind of random differential equation perturbation here, it's not clear that, you know, that's going to be invariant under reparameterization. And it's also not clear that after I've perturbed this, I still have a nice compactness result in the Gromov topology, right? So, yeah, you add some first-order operator to this, and who knows what might happen, right? So, the cool thing with sort of perturbing J is that no matter how you perturb J, it's always going to be a covariant, so this isomorphism group is always going to act on the zero set, and for every J, you have the Gromov compactification. So, it really only works when you perturb J, if you can get the transversality by perturbing J. So, let's see. Right, so I want to point out it's important to say always that this is actually an equivalent Fredholm section. So, that's this, right, so really what we would, you know, in our dreams we can generalize this to get equivariant transversality while preserving compactness, right? And that's just simply wrong. There are sort of two-dimensional counter-examples to that. Or here in our dreams we can somehow get, you know, transverse perturbations in each of the little charts fit together under the transition maps. But the transition maps are merely somehow continuous maps that factor through this totally irregular space, so it's not at all clear how you would iteratively construct that. So, we simply cannot, you know, once you stare at this situation, long enough you realize that it's really a little bit of a fundamentalist religion to believe that that theorem should have a straightforward generalization to J curves, right? So, general M bar, we can't expect to be regularized along the finite-dimensional regularization approach. Can I ask why equivariance was so important here? Why what? The equivariance. Right, so my compact-modulized space, what I expect to have a fundamental class is really this, right? It's a zero-set modulo, a finite-dimensional group, and then compactified, right? So, if I now perturb this by something to make a transverse fine, then I have a manifold here, but it doesn't even act on it anymore, right? So, this is not gone there, and so... Less zeros then? I mean, it's like more spot you perturb, you get the homology of the more spot critical. No, this has nothing to do with more spot. Yeah, yeah, it doesn't. I mean, so, see, I would want to write, so I need to write down a prescription for how to get from this to a fundamental class. And I would like to write down a prescription that gives me the expected thing if the unperturbed one is by itself already a manifold, right? So, if the unperturbed one is a manifold, then it is d bar inverse of zero mod this group and then compactified is a manifold. I'm taking its fundamental class, right? So, now, if this isn't transverse, I'm supposed to okay, you know, add a perturbation, but if I now simply forget this quotient, I'm going up by like six dimensions, so I'm getting, even if this somehow magically was compact, I had a compactification, it would suddenly live in a different dimension. So, it's clearly not the right answer. So, the point is that there are some cases in which it's transverse. Right. Yes. Right. And so, whatever description, whatever definition of this regularization I write down, ought to generalize the transverse case, right? That should be one of my axioms for the fundamental class, right? So, if m bar is already a manifold, it ought to actually be the fundamental class. So, this is important, right? So, if any of you don't understand why we need equivalents, squeeze people. Sorry? The group lack on E and B, right? Yes. Yeah. Right. So, this is really... There was a question here. Yeah. Are you saying that by instead changing the J, the capital J, that the term that you get, like in the difference that makes it equivalent? Yes. Right. That's the beauty. No matter how I change J, the capital J, it's... The reprametrization group is always going to act on it. Right. Yes. So, if I understand correctly, the problem is if you perturb in some laboratory way, and when you act by the reframetrization, you get some different motivation. Right. Well, I'm solving a different PDE, right? In the Kevin Costello's treatment of gauge theory, you ended up with some similar problem where you're trying to truncate some energy, but then the gauge group action doesn't preserve that. And so, you construct some sort of sense in which it's preserved by some group action of the homotopy. Is that hopeless here? Yes. Yeah. Let me leave it at that and we can talk more. Yeah. Maybe... Right. So, the main thing is in gauge theory, the gauge group actually acts smoothly. And so, you can do some of the things I'm about to do, and then the reprametrization group does not act in any pleasant way. Right. So, we're in deep doo-doo. What are we going to do about that? Wow. I am in doo-doo here. Okay. All right. Let's see. Well, I mean, mainly this is to energize you why we're spending these two weeks on this. So, I guess we've already talked quite a bit about the geometric approach. Right. So, the intuitive way it doesn't go through, we need to do something. Right. And so, there was the geometric approach that you've seen this morning, which has pros, you know, it works. It does all this magic. So, in particular, it gives a covariant transversality and preserves compactness. Moreover, there's something really nice. Somehow, it has automatic coherence, which is no matter what J I take. So, if my boundary or perceived boundary of the modular space is this fiber product, right, and that comes from some kind of bubble tree or breaking compactification, then no matter how I perturb J, it's always going to be like that. Do you notice that terminal logical question? Geometric approach means only perturb J? It means, well, it really, for me, means find some geometric way of perturbing your equations that does these two things. And really, you know, you're going to need geometry in order to preserve equivalence and compactness. So, there are, you know, various aspects of this. Right. And maybe I should say, right, I mean, there's work by Chile but Monke, who, you know, makes sort of more general perturbations of J that, to me, falls squarely under geometric approach. You call it Hamiltonian perturbations. Right. Hamiltonian perturbations exactly. So, the corner of this is, it just doesn't always works, right? So, it requires some kind of geometric control of curves, namely some kind of injectivity. And, right, another side note here was that people were first thinking about closed curves and their loss of injectivity means you're actually dealing with a multiply-covered curve and then you get worried about its self-symmetries. Chris called it Automorphism Group. I'm going to call it Isotropy, I guess. Which means you get into this obby-fold world and this is why people got worried about how do you actually make the Euler class of an obby bundle. However, the, well, we're not going to talk about Isotropy this week in Joel and Mike course at all because that's really not the problem. So, lack of injectivity is the problem and if you, right, and if you still think that not somewhere injective always means multiple-covered, ask someone about the lantern example. So, for discs, you have holomorphic curves that are nowhere injective but also not multiply-covered. And so, it's really the, and for those transversality it's really impossible to achieve pretty much. So, it's not the self-symmetries that are the problem, it's the, really you need some kind of magic in order to get this going. So, right, and maybe, right, so really the paper to read for this I would say is Fleur Hofer Salomon. And of course the, yes, yeah. Right, yeah.  I'm only going to write things on where I believe that if you have questions about it I can answer them. So, ooh, and I'm going to write. That means I need to not write names on the board right now. So, well, so I could not answer any questions about Foucaia's work or Tian at all. I believe I can answer most questions about Siebert's version. So, and this is also the most readable of all these accounts. And because it was the most readable it also was the only paper in which mistakes were actually found, and that's why this is the one paper that didn't get published. However, I believe it's actually the one that's closest to being fixable. So, or at least I understand more of it than, well, anyways. So, so there are essentially two versions of this. Foucaia at all is somehow a Kuranishi approach, and these two do something that I would call, I think I've called obstruction bundle before, but I really think it's a stabilization approach, which both somehow look easy, but the con is they either aren't easy or they are easy, but they don't apply to jave curves. So there's a whole bunch of papers about abstract ways of if you have a section of this kind of structure then it has an Euler class, and I think those are all correct except I don't think any j modular space of j curves ever gets cut out by a section like that. So that's when this piece of criticism applies. So, however this stabilization here really I think is the right idea, and this is very close to sort of a more conceptually good definition of an Euler class rather than by perturbing. So the polyfold approach, I believe Siebert should have discovered polyfolds 20 years ago if he'd done things carefully at the point where mistakes happened. However, so the polyfold approach as of now talks a lot about perturbing, I believe one could see the approach rigorous by sort of using polyfolds and stabilizing and then there's less perturbations in the picture. So what happens here? We still are going to use this right idea except we're actually going to do it and dot every i. So the idea is to really get ourselves rigorously into the setting where we have just one section of one bundle. So that was the idea here. Really I would have liked to describe my modular space by just one section and then I would have liked to apply my regularization theorem and the question is what kind of bundle and what do I then mean by and that's where language is going to come in. So these are things we're going to need to define. The key is that these things have global smooth structure whereas the stabilization approach of Siebert only had local smooth structures and what else? Right, just like before I should assume that the zero set is convenient. I don't know the difference between a global smooth structure. So if you have a topological space you could give it local smooth structures by just putting charts like local homomorphisms to Rn and if you don't assume that the transition maps are smooth then you have local smooth structures but they don't match and that's exactly what happens in all these papers. There are smooth structures locally in which the operator is smooth and it's just don't match. Whereas probably there are a good number of compact metric spaces that you can't even find local homomorphisms to Rn. So once we have this set up as just one section of one bundle there is the rough form of the theorem that hopefully we will not prove but now let me state this in a similar way as the regularization theorem up there. So pretty much we're just going to need to define things here in such a way that A they apply to homomorphic curves and B I have a regularization theorem. That's the goal here. And this is the regularization theorem so I'm going to write some set of perturbations roughly speaking this will be compact perturbations supported near the zero set with some measure. And for that non-empty set of perturbations again I have two properties for all p I have some notion of transversality and I still have compactness for all pairs I have a cobaltism right what should I oops right so now I need to cheat. So I am going to only talk about the M polyfold version of things and when this is the M polyfold version then these two here imply that the zero set is in fact a finite dimensional manifold compact. It might have boundary and corners but then now I could say what it means for two zero sets to be co-bordant right so pretty much exactly the same theorem except now generalized to some notion of bundle which hopefully applies to J curves and so the pro is that this seems to be a very clean way of actually getting regularized, modularized spaces and the obvious con is that there are words here that you are going to need to know so you need to there is some language overhead but that is why you are here so yeah. Just so we have an example or two to keep in our heads can you give like a sort of minimal example of something where you need polyfolds not to do much of a project. Whoa yes let's see right so if you have if you have a spherical Grom of Witton and Varen and there is negative Schoen class spheres then Dusar explained why multiple covers will no matter what J you have will always give you things that are cut out non-transversely so also if you kind of don't want to worry about the moment you don't know whether your curves are somewhat injective really but sometimes you can use domain dependent right yes I call that somewhat injective right so okay right so the moment there is bubbling so soon as there is bubbling you can't actually use domain dependent almost complex structures anymore because because the bubble is not of your original domain so on the bubble somehow automatically J is not domain dependent and so if you can't exclude say sphere bubbling you're forced to deal with well the question of somewhere injectivity you can use a Chilean but that's more sophisticated maybe right but then you can only do that if you don't have boundaries and it's not clear that you can I don't think they can do 50 so right goodness I am totally alright core points hold on yes oh yes this is hope of results gets and yes we'll have complete references by the end of this week so the other con that is the coherence so that the boundary of one perturbed thing is still some kind of fiber product of these modular spaces is no longer automatic so that is going to require some hands-on construction so so the coherence of this identification is no different right yes right so pretty much right so this kind of says that the perturbation on the boundary of my polyfold should be so that the product of other perturbations but the thing is when you're just allowed to somehow construct a perturbation just like a usual function it doesn't mean that on some stratum it's given by the product of some lower strata somehow that's it that's something you need to just do hands-on and that means you come out need to define these perturbations sort of iteratively complexity by complexity upwards and sometimes that's possible and sometimes it's actually simply not possible and then you need to sweat a little more so interesting things happen so so that's the master approach right and so evidently right now I need to explain to you how this fits in with what we think is actually the moduli space right we know how to describe the moduli space as a zero set mod something and then compact define we don't know how to describe it as a zero set of something right so so there's really a total paradigm shift happening here in that from describing the moduli space as this just one zero set right what happens here is I first perturb j then I quotient and then I sort of add in bubbles breaking and the like so we realize I can't get equivalent transversality so the perturbation somehow needs to happen after the quotienting and it actually also needs to happen after adding bubbles and breaking so the order in polyfold is that we first quotient then add bubbles and breaking and eventually we perturb so but that means there's something weird going on right so on this side a bundle I have my D bar operator and the whole thing is equivalent under my isomorphism group and so on this side what I should sort of do is I should divide first hope that that gives me a bundle hope that somehow my section still makes sense but not just that I kind of need to actually sort of add the broken things before I write my section and now I don't take this overline I should maybe make it a wiggly line so this is not a closure in any topology because those are never going to be compact they're going to be very infinite dimensional this is a function space though but I need to add all the stuff that I eventually want in my compactified modulized space before I solve the equation do you have the numbers correct on the right side I mean we don't divide up yes we do well in my interpretation of what you do you do so let me say what I yeah but nothing is still not correct you're adding a bubble and then you're putting out no no no but never mind so so what happens here this is the core ideas of poly fault theory so my theory has always been that if I explain to some reasonably smart graduate student these two ideas then I can lock you up in a dark basement for a year and you're going to come up with poly fault theory so let's see what do we need so first of all we need a smooth structure on a space of maps modulo the isomorphism group of my surface despite the fact that this action is pretty much never differentiable and that's what you'll hear from general this afternoon why this is never differentiable so pre-composition with maps is really really bad so this is pretty much so the only ways in which this is differentiable is either your isomorphism group is actually finite which doesn't hold if sigma j is for example 2 or it's r times s1 with standard j or a torus all of these will not have finite isotropy groups isomorphism groups and the other way is if map if this is actually a smooth if this is a finite dimensional manifold that sits in the smooth maps then it's also classically differentiable otherwise I'd challenge you to find any reasonable frichet space topology on an infinite dimensional space of maps in which this is at even one point differentiable is this why algebraic geometry is easier because in that case I can put map in exactly this is exactly why algebraic geometry is easier one of the reasons because I don't know because you don't need to quotient because instead of this you just take curves in m maps is finite now maps lives inside infinity and finite dimension because your analytic maps right yes yes maybe right analytic maps might be right and so the idea is here whoops so the idea is define so the idea is to pretty much define a notion of scale smoothness and prove that this action is scale smooth which then gives me a scale smooth structure on this space so in that sense I would say we quotient out first that's not what you wrote because you guys look at quotienting at last I just want to quotient okay goodness I'm glad we talked about it I'm glad we talked about it good okay boy do I have should be done you have two minutes good so this is a reasonable idea to have the second idea is really not reasonable that sort of so I ought to describe the neighborhood of a broken curve or a nodal curve as if it was in a Banach manifold right so somehow I want to revert to Joel's example right two spheres requires somehow maps from S2 to M times maps from S2 to M that's the data I need here for my Banach manifold but somehow nearby I'm going to have smooth curves and really here I just need once maps from S2 to M and this is a different Banach manifold from that in particular really the neighborhood is described by something called pre-gluing and that should become a chart map of the poly fold the underlying poly fold or M poly fold B so which fortunately also gets me out of the problem of fix of matching notation with Joel's because I'm not going to give you notation so what happens here you're going to somehow describe nearby curves by a pre-gluing map so I'm going to take so how do I describe all the curves so let's see this is now a point in my big base space and nearby there are some curves that are still nodal but I've perturbed both maps but there's also non-nodal curves nearby so how do I describe them I actually there's a rotation parameter which I'm not going to reduce and then there's going to be some gluing parameters which could be infinity and that's not good let me forget about the S1 here for a second so right so I have a nearby map and another nearby map and some gluing parameter and I'm going to make a pre-glued curve nearby which I get essentially from writing V1 with a long neck and V2 with a long neck and then I'm going to forget V2 beyond here and V1 beyond here and I'm going to interpolate on this little strip and glue the two curves together so it's evidently not going to be geolomorphic anymore but this is actually the answer to what is the topology so this tells you what a neighborhood is you do have a rotation parameter right in this case there should be a rotation parameter indeed yes which I just end of lecture great good so the key observation here is that this map is supposed to be a chart map but it is clearly not injective because I'm forgetting data here and so the idea is to say well it factors through something that is a really weird subset here but then it factors through something where I have a local homeo here and this here is something like a retraction so it's a subset in here and I have a controlled way of getting from a general point here into this set and that's really the key to happiness here is because while this is a weird space it has an ambient topology so this ambient topology is matrisable and it has a smooth structure and everything is good unlike this microphone so I shall stop here we have questions for Catherine you mentioned before a big philosophical difference between this idea and the kind of abstract perturbation that I sketched right because what you sketched sort of said I perturb here right but then I have to preserve equivalence and compactness so the point is the first step here is figuring out how to describe the whole modular space literally as locally the zero section of something correct yes right which means I need to figure out what my ambient space is right I want an ambient space for my modular space that the compact modular space actually sits in and that's actually that's sort of a general problem that's exactly that problem the problem of an ambient space is what you end up having to deal with with Kuranishi structures what you also end up having to deal with the Tian and Siebert approach it's always what is the right ambient space in which I could describe things as just one zero set yeah so what exactly do we expect from smoothness in this picture smooth some kind of implicit function yes okay then but then in the end what we use is this classification of one-dimensional manifolds right oh if you do one-dimensional manifolds yes but but even a one-dimensional manifold in order to get something from a Fredholm section that's one-dimensional you need an implicit function theorem right you need a notion of transversality so you probably get away without smoothness I would say maybe C1 just for the implicit function theorem but it's going to get technical now so sort-smale is the reason I believe you really need smoothness and not CK why is why do you think scale smoothness is such an obvious idea that a graduate student could come up with it I refer to Joel I don't think that's what she claimed I think if you explain to a student my claim was if you give somebody the definition of scale smoothness and the definition of what these retracts are then yes that's not quite what you said no I know well my graduate students of course alright when did you get rid of the F1 parameter in the end why did I hmm I didn't want to have to deal with trying to explain the rotation because also technically I shouldn't write S1 times this because at infinity there's no S1 parameter so what I do at infinity here is I just don't glue and when I don't glue I don't have an S1 choice and that in fact is the reason why this is not a boundary point it's completely an interior point so you're going to see I think tomorrow Joel is going to explain to you that really for nodal curves really the gluing parameters are just a disc and where the middle point is infinity so so here pretty much if you think of these as discs then you don't have an S1 parameter and then you see where the boundary comes from but when there is an S1 parameter then infinity should come without that and we'll have anyone with like philosophical questions of like why can't you just I'm most happy to talk to you all week and in particular maybe we'll have a session this evening where we think about why can't we just just thank you