Do you want to start a cloud kitchen aka a ghost kitchen? Well, today you're in luck because I'm going to be showing you the steps to create your business plan So you can create a thriving cloud kitchen. Make sure you guys keep watching Hello friends, my name is Wilson your friend in helping you build a thriving small business and a profitable food business Are you thinking about starting a cloud kitchen? Let me know in the comment section below and by now you have already heard of all the raves about cloud kitchen virtual kitchen and Ghost kitchens and if you haven't already make sure you guys check out this video where I talk about What this whole thing is well for those of you that don't know what cloud kitchen is cloud kitchen is like Airbnb But for the kitchen space, what does that mean? That means you don't need to do renovations You don't need to hire a bunch of people and on top of that. You don't need to sign a three-year lease That's the beauty about cloud kitchen. It is much cheaper to start than a traditional restaurant Now having said this that doesn't mean you should skip out on all the right fundamentals to start off your Operations the right way and that's the reason why you must start your operations by writing a proper business plan So first and foremost, why do you need a business plan? It is because it provides the clarity for your business clarity for your vision the fundamentals and all the different areas of your business That you need to think about for example Marketing your recipe your pricing all these things needs to be thought about in the very beginning So then that way we know exactly how to get to our destination when you're running a business It is very easy for us to lose track on what we're doing and getting confused into the mixes And that's the reason why having a business plan gives you so much more clarity because you can always come back to this As a reference point about now it is time for us to work on to the next step Now the second reason of why having a business plan is so important is because it allows you to find the right partners Now what do I mean by that because everything that we think about the vision that we have are all in our minds We must be able to extract what's in here put it on paper and to actually go and find the right partners When you are able to show the vision that you have to potential partners now You can attract a listed players and partners to help you out to partner up with you to build this big brand That you have in mind and if you're thinking about well, how do I find the right business partner? Then definitely check out this video where I dive deep about finding the right business partner And lastly majority of the people that are out there to create their business plan is because they need to raise funds They need to raise funds from either friends families angel investors or the banks now Why do they all ask to see a business plan? It is because they need to see you Are understanding all the fundamentals of your business. They want to make sure they get their money back So the more detailed and the more areas that you cover The better it is for you to get funding for your ghost kitchen operations Now as a disclaimer I'm only going to be covering all the essential parts of a business plan So then that way you can understand why these points are so important Now before we dive into the 10 critical components you need for your cloud kitchen business plan Such as your team your financials your concept marketing Go ahead and smash like button because that's what's going to get you the funding You're going to get the funding from your friends your family the government Smash like button so your dreams will come true. All right.  What is your concept?  Write down your concept and your unique selling proposition. What is your unique selling proposition? Why do you exist? What makes you special? Now when we're thinking about this concept, we must always come back to the problem that we're solving What problem are we solving in the marketplace? Are we serving comfort food? Are we serving a really good delicious bites for share during game time? We must identify the problem we're solving and the unique way of why and how we're bringing it to the marketplace That's what makes you special and the last point when it comes to concept You must build something serve something that has a proven demand For example, one of the biggest companies out there fazus They only create brands and concepts that has a proven demand in that specific area And that's the reason why you must do your research You must study what your competitors are doing to get inspiration from them to serve Something that you are serving that is uniquely you now We don't want to sell something that does not have a proven demand So that's the reason why always do your research Now if you are having troubles finding the concept idea finding what you want to sell finding what is in demand And what is popular check out this video where we talk about the most profitable items for a cloud kitchen The second point that you need in your cloud kitchen business plan is your team You must include the credentials of your founding team. Why is that important? It is because I want to know as an investor who are the brains behind this operation What are some of their technical background that makes them have an edge an advantage Of running this business, especially when it comes to ghost kitchens We don't need you to have a lot of education. No, it's not about the education because ghost kitchens is heavily on Operations, so if you can highlight some of the examples some of the areas that you have Done in the past experience that you have from the past that can highlight this specific area Operations, then you're going to have an advantage over other people who hasn't done this before Another area that you need to add into this Biography of yours is any credentials that you may have any other resources and networks that gives you an advantage Overall, we want to highlight what makes you special What gives you the winning edge over someone else Who's operating this business Now a lot of people have been asking me what if I just don't have any experience running a food and beverage Location it is completely okay because we just want to highlight what makes you special What makes you unique what makes you deserving of this opportunity? So if you cannot think about anything Then you must go back to the drawing board You must be able to extract something that makes you special makes you The one that is deserving of this opportunity And lastly when you are assembling your team, you must look at what each individual Brings to the table. What value do they bring to the table that other people don't have? For example, when we assembled our team for our ice cream shop I brought in my supplier as a partner and the reason why I brought him in is very Strategical it is because he can provide us with the lowest rates We brought in an operational specialist who can run our locations for us And that's the reason why I brought in this team and assembled this team for very specific reason Which ended up allowing us to sell this whole business So for you assemble your team strategically and see what they bring to the table Now the third component for your business plan is target market So remember I talked about how your concept solves a problem in the marketplace for a very specific demographic Well, this is where we highlight this problem We highlight the problem the specific person is having and how your product Is solving that problem now For example, if you want to serve the young millennials looking for some hip food looking for some comfort food And you're serving smashed burgers then the branding that you person must be more hip must be more millennial Must be more modern so then that way you can attract this specific demographic By you being able to clearly identify who you're serving Now you can understand how to communicate with them where to communicate with them and what to communicate with them What are they struggling with what are the biggest frustration? Is it not being able to find the greasiest most comfort food at a convenient location at an affordable price? Now if that is a big problem, then you can solve them and meet them where they are at Now when it comes to marketing. Well, where are they hanging out? Is it on tiktok? Is it on instagram? And if they're on instagram or if they're on tiktok Then you are going to do all your marketing on those platforms because That's where they're hanging out by you clearly being able to identify your target market You can now profile them that allows you to not only market to them, but also to sell to them properly This is a very critical component in your business plan because this gives your partners your investors and yourself The clarity of the problem you're solving for your customers And if you're able to match that up properly, then you have a thriving business But if you fail to do this, then you should go back to the drawing board and find out What is the problem you're solving and whether your product can really solve this problem for this specific demographic And if you want to increase your chances of getting even more funding from your investors or your friends or your family Then you must attend this free live masterclass in the link below This masterclass is where we cover on how you can create your food concept and make it a thriving business Allows you to understand how you come up with that idea How you identify your customer avatar and how you can market to them properly So definitely go sign up for this free masterclass in the link below. I'll see you guys in the class The fourth component for your business plan is your location location location location We always emphasize the importance of location And this is no different even when it comes to having a ghost kitchen operations This is also one of your biggest startup costs So you must be able to understand how to pick the right location And the variables to look for when choosing your location Is the location at where your target market lives if you're selling ethnic food Is your location going to serve this type of people? Does the location have the equipment that allows you to create your type of food? What are the hours of operations? Are they flexible? Is the location easy for drivers to come and grab your order to deliver?  Is the location safe and secure? So then that way you don't run the risk of being robbed because once again when you're in this trade You're going to be here for hours on end. Does the location have cleaning and disposable facilities? Does the location have city permits and do they abide to the regulations? Do they have good inspection history and what other additional amenities do they have? Now these are just some example questions that you should ask when finding a location So definitely do your research understand all these type of questions So then that way you can make the best judgment The fifth component in your business plan is your sample menu. What are you planning to serve and why? Every item that you serve on your menu should have a very specific reason of why You should always challenge the menu item that you're proposing With why and what problem does this solve for your customer? Are the items good for delivery? If you're selling fries probably not the best idea because it takes 30 minutes to get your customer's place And it's going to arrive all soggy Are the menu items that you're proposing comforting because a lot of people that order food from home They are looking for comforting food Are the items that you're offering going to still remain tasty after 30 minutes? Or is it just going to be a clump of a big mess like for example spaghetti? Are the menu items you're proposing complementary items making sure that it upholds your hero item? Are these items supportive of what you're serving for example, if you're selling burgers Complementary items would be drinks would be desserts would also be fries because these are the items that you can include as a bundle That makes your average order value much higher Are the items you're proposing giving you good margins? Do you understand the cost of goods sold? How much do you actually make by serving each of these items? Now these are just some of the questions that you must ask yourself when creating your sample menu Now you might be thinking well, Wilson, I really don't know what my customers are wanting The best way you can do this is to go and actually survey them ask them what they're wanting Ask them what items they would order in addition to a burger Ask them how much would they pay for it? The best way for you to find answers is to validate it through your customer avatar your target markets The sixth critical component for your business plan is your design. What does your branding look like? What does your logo look like? What is Your whole experience for your customer. Are you going to customize your packaging or do you not care about your packaging? How does your logo look like? How does your instagram look like? We must be able to share our vision what we have in our minds with our investors with our partners So the best way for us to do that is by designs by us including these mock designs in our proposal Now because you don't have a brand yet, or if you have problems finding a designer There are multiple different resource where you can find a great designer resources online such as upwork.com Or fiverr.com the seventh component for your business plan is the SWOT analysis Now it is great for us to identify who we're serving to our branding What our concept is and our menu. These are all great things But without us understanding what is in the marketplace. It is very difficult for us to maneuver We must know what is going on and that's the reason why it is so important for us to create a SWOT analysis What does it stand for strength weakness opportunity and threat Strength what strengths do you have over your competitors? Is it your branding? Is it your unique way of your recipe? You must identify what makes you special Second weakness. What makes you weak compared to other people? Is it your lack of experience? Is it your lack of resources? What is your weakness third opportunity? What is the trend or the gap within the marketplace that you are trying to take advantage of? This could be in price point. This could be in convenience What is it in the marketplace where you see an opportunity that you can fill? And lastly threat what puts your business in a very vulnerable position whether it be increasing rent prices Or increasing Ingredient prices or your competitors. These are all posing threats for your business by you identifying and creating this SWOT analysis This allows you and also your investors and also your partners to have a clear vision of what the marketplace Looks like so you and your partners are most informed in this playing field The eighth critical component for your business plan is marketing This is by far my favorite section because this is where all the sexy stuff happens How do you plan of bringing your brand to the marketplace because at dinner date? It's great to have great food good quality great photos But if you don't have a plan of action plan of attack to go to the marketplace to let everyone know About your great creations. You basically don't exist being just on third party delivery apps is not good enough We must identify what are the steps? What are the plans? What are the strategies that you have in mind to go to the marketplace? Some examples of marketing that you can actually use for your ghost kitchen would be running facebook ads social media influencers collaborations brand deals Contest seo your website email marketing a lot of different areas that you can do Choose one or two or even three for you to focus on have a detailed plan So then that way you can execute it to the tee once again It's not about having touch on all these marketing channels No, it's about being good and having deep in these verticals choose one to three So then that way you can really execute well Now if you're still in the idea phase Then you must attend the free master class in the link below this master class covers an hour long of business plans ideas Marketing and also how you can actually bring an idea to fruition to marketplace create a good thriving food business In this free master class. This is what we cover go sign up right now And i'll see you guys on the inside next up We have financials great to know the marketplace great to know what you're selling But if you don't know your numbers you're in big doo doo, so you must know your numbers.  How much are you investing in building this whole business? Everything should be as detailed and as accurate as possible Everything that you have in mind everything that goes into running a business You must include in this financials because the more you can put down the more accurate it is The more accurate it is the more feasible the plan becomes Last thing your investor wants to see is something that is not accurate something that is not realistic Then it shows the lack of preparation on your end and that means you're not ready for their investments Now when it comes to determining the startup costs, we must identify how much inventory we're budgeting What are the design costs? What are the licensing costs? What are the initial investment that we need to put for rental? How about labor? These are all key components that you must include in your startup costs So then that way we know how much we are asking for from the bank and how long does it take for us to pay back this investment And lastly guys, I want you to end off your business plan with the vision The vision of where you want to bring this operation this whole brand. What are you trying to build? What makes you want to jump out of bed every single day go through hoops go through hurdles Go through all the hardship Just to create something you're so passionate about Where do you want to bring this brand? Identify that ask yourself that and put this in the business plan because ultimately Facts will not sell You must sell your vision because that's what other people will buy into you. It's your vision So there you go friends the 10 critical components for your cloud kitchen business plan These steps you must go through them thoroughly because this is going to increase your chances of getting the funding Getting your partner of your dreams Getting to start something you're passionate about and if you want additional training You should definitely attend the free masterclass in the link below where we share with you some executable Strategies that you can include in your business plan, which would increase your chances of getting that investment This is a free masterclass that i've conducted for you. So definitely go check it out in the link below Otherwise, I really really hope that you enjoyed this video And if you do help me out, please and smash the like button. Otherwise, I'll see you guys in the next video


Everybody forced here ripping 2022. I'm act wwe hobby five bucks case number 15 random hit style 25 spots in the break There's 30 cards in the case. We're gonna combine the base parallels with some of the relics here If there's a base parallel one-of-one that's gonna be bigger than most of relics, so that'll stay on its own That does happen. That'd be cool. Pull one But so everybody is going to hit in this everyone will get a hit Good luck y'all here. We do we'll type in the hits as we go and random everything off at the very end. Good luck Hey, what's up? Wahoos? You know, who's can't miss the other ripping of an iMac case Yeah, but he's always good Churchill Churchill would be cool to fool. Yeah, they always have like 20 or so like presidents or like historical figures and they'll have like 20 or so like pop culture then they'll have like They're like at least half are like baseball really some somehow We'll have like astronauts Here we go, I'd say when we did Bowman we did pretty well in the cut autos though I will say I'm a transcendent hit like Garrick Jackie Clemente Abe Redemption a redemption always our redemption is almost like a good thing Hey Booker T my guy 60 of 65 base parallel Booker T. He had the glass the glass shattering Intro right and that like sound effect Boa relic 53 of 99 stand-up memorabilia is a Carmella relic 93 of 99. Yeah in the back. It's usually a really good thing Here's an auto of Tom Pritchard 14 of 99 Yeah, I'm gonna get that jacket Look at that jacket. Gotta get that. That's fire. Okay. Can we focus on the car now camera? Relic auto is ricochet. Yeah Vaughn. It's gonna be it's gonna be pretty nuts I'm actually really excited for opening day this year because all 30 teams are playing 15 baseball games on opening day It's gonna be beautiful if none of them get canceled or like po like postponed by rain is what I'm trying to say like like weather That'd be freakin awesome. That's ricochet And a redemption macula milestones the Hulkster Hulk Hogan Good luck on this one at the end everybody Hulk Hogan Immaculate milestones big-time redemption We like to see Vaughn are you most excited for a cluber and Kyle Gibson? I'm pretty excited for that This McLane hand against Hoover's is the race. It's pretty lopsided opening day for the race at least like pitching matchup wise Yeah, McLane has a beast Whoever they're playing it's like it's pretty lopsided It's like not even their best pitcher mankind base 18 of 65 It's weird how that works while who's Is crazy? Lacey Evans relic number to 70 one of 99. I like the international games, but I also like that I don't like it being opening day like opening day should just be on the ballpark seer Especially since we said the ball baseball classic shirt relic Gigi Dolan 6 of 50 Jumbo relic Gigi Dolan 6 of 50 Clanahan erotic. Yeah, Eddie aces. Yes. He pretty lopsided patch auto Reggie tenant 25 character from rocket power Reggie On the patch auto Reggie was the girl, right? Patch auto AJ Styles 87 of 99 Maximize Squid Auto Yeah, twister. That's Maurice AJ Styles there and signature moves Bianca Belair auto 7 of 49 Sick Bianca. Look on this one everybody. Yeah, sure is all contra. Yeah There's a Rhonda Rousey base 42 of 65 Shers are all contra I know sometimes Sherser has the rough first inning that settles in he'll give up like one or two runs and then he just cruises That's Rousey. It's actually first start of the year. Okay, get tricky What is the Marlins line-up son? Gunther relic 34 of 99 Gunther relic NXT stand and liver shirt relic here is for Raquel Rodriguez 40 of 50 That's what I'm saying Vaughan. It's like it's like 1 p.m. To like 1 a.m. Baseball. He's so good Raquel Rodriguez shirt 40 of 50 Relic Auto drew McIntyre. It's a nice relic coming up next your McIntyre 85 and 99 relic auto and Relic Auto 2 of 10 Carmella Very nice relic there sparkly Superstar worn Carmella relic auto and again a Bianca Belair relic auto 42 out of 99 really was I didn't realize it was that much Bianca Belair very nice. Yeah, the shirt relics are pretty sick in this. They're like the best like just base relics That are like a big-name canvas. I feel like would do pretty well Alba fire on the base parallel 10 of 40 a Zoe stark relic 24 of 99 so we stark Jumbo relic Alexa Bliss 82 of 99 a Becky Lynch to 19 Must have been pretty good night that in the triple. I don't know if you're watching the night Oh, I was about to pull the triple Dudley Dudley auto Redemption I've our auto 42 of 99 got a Shinsuke Nakamura 70 of 99 Nice Nakamura there and another nice relic auto is a Becky Lynch 85 of 99 Becky Lynch 85 out of 99 All right, we need something huge here to close out the case. There's been there's been consistently decent autos I want to see a big one here for People to hope for in the random other than the Hulk Mr. Perfect Kurt Henning base 13 of 40 Pardon me, Mr. Perfect never make a mistake Dolph Ziggler canvas 12 of 99 I'm standing the liver Rhonda Rousey canvas 37 of 99 Rhonda Rousey WrestleMania 38 night one Relic auto of Drew Gulak eight of 99 Herald and signatures the iron the iron chic 93 of 99 legend hit there Iron chic and on the back. We've got a legend all-time greats auto Trish Stratus 45 of 99 Good luck on that one Trish Stratus Is she a WrestleMania who's Wow? I'll have to watch some of the highlights for sure at least Very nice Trish Stratus Okay, let me combine hits here. So let's go Booker T can go with Carmella Okay, so not the Gigi Dull and not the Rick already has got it mankind base with the Lacey Evans relic is cool Rhonda Rousey and Gunther can go together Calvifier and Zoe Stark and Mr. Perfect and Dolph Ziggler. I think that works actually Dolph Ziggler. We'll go Mr. Perfect and Random time everybody best of luck. Wow. That sounds insane. Wow. Who's 12 times Got the long sweat here 12 times names and hit same amount of times. We'll put them side by side one two three four five six seven eight nine ten 11 and 12 Michael D down to Jason C and now the hits best of luck everybody one two three four five six seven eight nine ten 11 and 12 Drew Gulak and the Tom Pritchard. What was it Tom Pritchard again? So just has the name and right Michael D with the Gulak Damian T Bianca Belair relic auto Just not okay one second here Scott are with the Bianca Belair non relic auto Daniel and with the Raquel Rodriguez relic Ross see the whole Kogan redemption congrats Ross John Oh the bow relic and mr. Perfect base Kevin are Alexa bliss relic Sean are Shinsuke Nakamura relic auto John O Ricochet relic auto Earl S Becky Lynch auto Brian B. Dolph Ziggler relic Evan H AJ Styles relic auto April L with the Gunther relic and Rhonda Rousey base Scott are Ivar auto Derek R Lacey Evans relic and mankind base April L Reggie relic auto and Gigi Dolan relic Logan T with the Trish Stratus auto cash K with the iron chic auto Andrew J Zoe Stark relic and Albu fire base Dan F Ronda Rousey relic Damian T Carmella relic and Booker T base Tim Z drew McIntyre relic auto Erin M Carmella relic auto to 10 and Jason see the Tom Prichard auto get those posted. Thanks again everybody


So welcome to University of Greenwich, my name is Charlie Pilvin Bailey and I am the Programme Leader for Forensics here. So I want to talk to you about our new programme which is Forensic and Digital Investigation. Now this means you get the key expertise from two different subject specialists so you get from our Forensics team down here at Medway who will teach you more about the crime scene related work and laboratory work and then you get the digital expertise from our Greenwich team who have a wealth of experience in digital investigations so looking at things like embedded images, steganographic techniques. So learning how you can manipulate digital data to find out where these items have come from. This is hugely important in terms of your skills because it impacts on so many different areas of everyday life from the general public who we serve as forensic scientists. So this covers everything from if you are using your phone for incident banking or if you are using RFID chips in your contactless card payments. So these are areas where people's identities may potentially be stolen. Digital investigation links into all areas of police and criminal investigations so including forensic investigations linked to missing persons it links to the black market it links to fraudulent transactions. So there are so many areas in which we are so heavily reliant as a society on digital data that it comes a priority for us as forensic scientists to investigate those areas and look at also prevention methods. This is hugely important for us and it's hugely important to the general public that we do everything that we can to try and find a resolution for this. So here at Greenwich we work with you as part of your personal tutors so the idea is almost like a mentor. It means that if you're having a rough day you can come and talk to us and it doesn't have to just be about your studies it can also be about anything else that's going on but the idea is that you can chat to us and we'll help you through those hurdles. As part of your digital and forensic investigation you will be looking at all sorts of different areas so you will need to use specialist equipment when you are taking this data so what you will learn as parts of forensic scientists is that when it comes to digital data you don't work off the originals because as soon as you start manipulating them you are changing them.


Okay, it's something it's time for Alice to talk about package in KV Bradley KV because it's All right, okay, so well first of all let me introduce myself a little bit For those who just know that I am the guy who talks about the slim book. My name is Speak up Yeah, if you cannot hear me just wave like that. I will take it as a sign My name is Alice Paul. I come from Barcelona and I've been developing KV for for a while now actually this February It's ten years I work for Blue Systems where we do KD stuff and Well other things as well, but mostly KD KD software and Things around KD these trolls and so on and I'm also part of the KTV board if anybody has a question about KTV things we can discuss them later outside if you're interested and One of the things I've been working on on my free time because I also do KD on my free time like my life is about Katie nowadays is Flatback and snappy tooling and things and That's what I wanted to talk about today. So before getting into the Into the subject Head start I wanted to talk about a bit the motivation like why would KD be interested in These systems, maybe it's something obvious. Maybe it's not so before I say anything Who thinks here that it's obvious that Katie should be doing? what Container stuff Okay, so for the record, it's about 20% of the room We've been doing software in KD for 20 years now and In KD we have a vocation for The end user what we do in KD is we make software so that people will be using them on their laptops And we'll be using them or their phones on some occasions and and so on Um Historically we've done that through mostly through the distribution kind of system where we basically publish Turbo file with full of text files most of them well C++ source code but not exclusively of course and Images and a lot of the stuff and then there's other people who build it those That's only for Linux. In fact in KD we do other things as well in KD we have some of our software works on Windows some of our software works on OS X and on Android and In some occasions some even more marginal architectures and And we've done all of that work because or One of the important things for us is to make free software for people to use in their daily lives and to be able to to enjoy the tools and From my point of view, which is not a widely shared one We've had a problem there which has been that sometimes our users haven't had the The our latest developments and this has been a bad thing this has been a bad thing because Whenever we release a software we consider that stable right and if there's a problem with that software What we do is we fix it and we make another release if the users are not getting this new release They are these users are what handicapped on on on our software and It's really frustrating for for our developers to not be able to fix the issues and that's that that's something that This these systems try to solve so I was telling me now that he's not super sure About what these systems do so I will try to summarize it a little bit Both flatback and snappy are well essentially very similar They do have some differences, but in general what the opportunity they offer is to be able to distribute Applications like compiled applications that people will be able to use both of them offered the possibility to execute those applications in a sandbox environment which improves the the security and facilities for for the user to to adopt those systems and or those applications and Well, that's more or less the idea Is there somebody who still thinks he doesn't understand what they do I know my Definition was not great, but Well if there's doubts I can try harder Well The difference is that they're developed by different people at different moments in their lives, so well While they were trying to solve the same problem somehow. Well, they had different priorities at their moments and different things From a very high level point of view, I would say that flatback is far more focused to desktop applications that's Difference your difference is that the model of this distribution snappy has a centralized repository by like by Canonical or Ubuntu, it's called store dot Ubuntu dot com and you will fetch from there Flatback expects everyone to have their own repositories or at least it has been like that for a while They also have thoughts of centralizing it Another very interesting thing from us that we're looking at both is that they are kind of converging into doing the same things like one started from One end and the other started from the other and they were both very convinced But then you see how they start to Converts and go the similar Directions, I think it's it's important to understand when talking about this is that these technologies are not super mature They're ready to be used and and they do things like we will see hands-on how how it's it's working today already but that doesn't mean that in One two years, there will be exactly the same as they are right now already Flatback for example, they had an LTS release Some some weeks ago But I am confident that we will see still some development and improving on the on the platforms together But yeah, well, that's more or less What I wanted to say so in general for us is to have the possibility to like when we have a new release of the application to be able to Put it on on on our user hands one one example that that we use to explain this in K develop is that Many of our users for example Our own universities universities the systems over there are managed by system administrators and One thing I've always heard from these kind of people is that they tell us well We're using these these software from five to ten years ago, which I understand it's it makes sense for for Well these systems to use Very stable basis or well, I wouldn't call them stable, but well, it's the Common mind share nowadays that all software means table, right? but For us when we make a new release with new features We want to tell these people well use the new thing we are doing right because we actually make fixes And we actually work on new useful features, but if these people they cannot use this software then Well, we still have some slightly unhappy users that We won't be able to to make happy up until five years in the future when the sysad means decide to brawl new versions of everything and when they do they will do it on All software at that point and it makes sense like for kernels it makes sense for even desktop environments possibly maybe Actually, I don't really think so, but maybe But it doesn't make sense for K develop, right? It doesn't make sense for Kate. It doesn't make sense for LibreOffice I would say if the developers say this release is stable then this release is a stable one and If somebody disagrees we can have a conversation on the question rounds later, I think it will be interesting But ask us Katie. What have we done? So how does that materialize? I guess that if you asked a known person They will tell you well, we've developed most of flatback Together with breath hat so but that will be their answer. We haven't developed flatback. We haven't developed a snap Although some of our developers will have patches on either platform But what we have done is we start to be on both platforms because we also see that there's none of them That is 100% what we want or that we think that Well, we all will have users on both platforms right so we need to be on both in any case So it's still not up to us to decide what Where to be So we've mostly been packaging applications that doesn't mean it's been exclusively applications, but it's been mostly applications applications is so If you are on wound to normal unity, you will be able to install Kate on on it And it will make sense if you're using Fedora, you will be able to also use Kate on it and The one coming from us, which is the one we are Well on top of So now I will go through All aspects of both and with what we've done on it I will start with how would you install applications then I will go through how to build applications Katie applications on those platforms, and then I will show you how we are integrating those platforms into our software center I hope that's interesting for you. I think it's really cool, but then I've done most of the work. So I'm not really representative How do in to install a snap well, there's two things that you want to install And Katie with we have this thing which is our frameworks We also all of our applications rely on cute, which is a fairly big package. So we needed to At least share of one of those Many of the complaints that I hear whenever somebody talks to me about bundled applications is how Every application is is well packaging all of all of the libraries and Well, it's certainly something we didn't want We don't want every Katie maintainer to have to maintain the Compilations of all cute and of all Katie frameworks. That's definitely something crazy And of course every time somebody downloads an application We don't want them to download all of cute and all of Katie frameworks because that's well Tense to hundreds of megabytes. That's not acceptable But then for us we consider it's acceptable to have a big package Which would be the frameworks package which has all of cute it has all of the Katie frameworks and then have Tense to hundreds of smaller applications To that you can download and enjoy and then have them all sharing this this big framework's package so If you want to read more about that there's Harald's block which explains a bit how it's installed Do we want to install our application? And the same for like if you have a dependency on 5.8 Like it is a version or is it like Well general whenever there's a new version Yes, the question is Do we have to have installed many frameworks packages all the same time and I would say no no well in both in cute and Katie frameworks we We offer background ebi compatibility So any newer package would would work with an older one which shouldn't be a problem for us There's of course the possibility to do it. It's not something. We're very interested actually Our interest is that people only has this one and to have to maintain this one, right? so and We don't want to have to maintain different versions of of frameworks or anything like that like Have to review it when SSL changes or whatever So yeah, that's one thing And then to build those maps I will go through More or less what the manifest file looks like I will try to go very fast because I don't think it's super interesting but I still want to to look into it first I want to show you the repository or I've lost the mouse the repository we have for Manifest files so the idea is that while Not all of the applications are here because in KD. We have more than three those three we want Application maintainers to maintain their manifest file somehow So that they get to decide when they want a new dependency or they don't want it They get to modify it and have the application compile like that We are still beginning with we're first starting and actually what we're doing nowadays is Using the neon packaging for creating the snaps, which is basically Magic how I worked on and it allows us to have virtually any KD application available Like right now not in some months, but now which is something we welcome So that's more or less how it works So on the manifest file what we specify is Well names and how we want the application to be confined So the information we have on the manifest is more or less the information that Snap the snap system will have of our application when it once it's running, right? So there's name version summary is for because snap is also at the moment System snappy uses for for listing applications and showing them on a software center also We need to tell it what kind of things our snap will be doing to communicate with the external world, right? This is those plugs If I say something really wrong, tell me I am not an expert but the idea is that You request these plugs and then the system provides you with the other things. So it's a modular system, but there's snap Developers over here. So what you can find them and then ask them these questions Also, Harrell will be able to answer you these questions better, but more or less here. The idea is that Here for example, we're saying we need unity seven and it says here notifications Well, what Harrell meant when he brought that is that you need to have somebody listening to you on that end So when you things Well, these notifications happen on the on the system side, right? because if it's confined and Nobody gets to listen to your application then you can send notifications, but they don't want show on on the on the system, right? and then That's exposing the KD frameworks package on So when our application is compiled is using the the shared frameworks thing on the On the build time Well, no, we want distribute unity with every KD application It's just the name that they used for for naming some interfaces and Think about it as a permission system You're basically requesting permission to talk to unity seven. That's what this line is basically saying What it's really doing is opening up a debus interface So Harrell suggested for the microphone that it's you can think of it as a permission system genuine question I don't see that here So there's this part which Harrell can you explain it? You will do it better come here And to quickly answer the question the way you would do it right now is probably through a different package name so you You've seen how the Thing is called kitty frameworks dash five right so if you wanted a version it would be dash five point six Right now. I don't think you can actually look it on versions There is no way not right now anyway So you would have to do this way through through the separate package, but beyond that there's no way Well, essentially the idea would be that you will always have the latest frameworks version, right? What's the what's the point of compiling against an older version? Yeah, but if you see it failing then you will change your channel into a number one and then it should work There's another question I think you should really ask canonical people to give you a snappy presentation I Think it would be great Well for what's it worth. I'm not a flatback expert either so Maybe this will be a horrible presentation Anyways What we do on on flat back you will see it's it's fairly similar Like I said on on snappy the the repositories are centralized That's why you just install things here. We need to add the repositories with with things as well What we see here is that we are adding a KD remote then adding installing the platform file from it and then You add another remote which is the repository we have for applications And then we install an application which in this case is ocula Yeah, we have a wiki you can look at Wow and Well, you will see all of the information there so As as you can see here, we have more or less the same structure we had on on snap We have the the platform with all of cute and KD frameworks and some integration packages And then we have the application elsewhere with only the what the application itself and other libraries that will not be on the on the platform like Ocula in this case. It has poplar, which is the library. They used to render PDF files and so on so Also flatback has a technology to talk to the outside Beyond the similar system that Snap has with the with the plugs It also has the portals, which basically are standard interfaces that are present on flatback as a platform so that you can request things So for example, the normal Open file dialogue you will instead of implementing it inside of the application you and requests for full file system access what you do is you just make a flatback call to the flatback Portal and then product will do the file navigation with all of the Privileges available and then give you the file somehow and Well, we do have some implementation of it You can look at this blog later and you will see what's what and then Well, the common to build an application is this one in in in flatback You need to install the SDK, which is basically the header libraries that are required and while some compiler stuff and Then run this beautiful line that didn't fit on the screen This is what Manifest file looks like for flatback where you basically what you say, which is your runtime which are the things you need from the outside and then the libraries you might need and then the application and Then you compile it now. It might be a good moment to look at an application opened on those platforms Let's let's see if I can do it without Without messing up so much So we can So the Applications repository we have for flatback is actually quite new. I uploaded it for the first time on Thursday But we've been working on the applications for a while these are the applications we have packets nowadays We can install one what we need to tell it the Repository and now of course you need to install it over it's already installed. You can believe it works. I Didn't make flatback though and that's ocular being run in flatback This is not running the The the portals yet, it's just Requesting the access to the full file system, but then you can see if we go up here There is an app directory which is actually where the application is mounted So this is not the root file system of my system. It's just the mounted system on on on flatback On snappy Let's see what snaps I have installed because we don't have a lot of time We can run scumbium who doesn't love scumbium boom oops Nanos gonna work Oh, yeah, there it is Yeah, no, that's not super I should have a KD application which one K blocks K blocks Sudo, right? So the snap Insulkey blocks. This is another application single game and then I could can play here. It's awesome now we can Then We can look at the software center integration We look first that discover because it's the one I did. I will open it with just the snap back end because well It's a bit easier. I would could have installed the application now. Which one which application we have as well. We have ocula ocula Ocula We get to open it install And we wait We don't have progress. That's great Well, while it insults We can because we don't have a lot of time. We can look at the video Jan from Red Hat did with the integration for on flatback for discover for flatback You can see it works. It's on a on a brand and it's actually not compatible with master. So I don't really want to install it now but you can see listing works and You get to launch the application how's that and Well, let's see if I Killa where is installed or I still have console Accuracy to be recompiled. So that's that's it There you go. So I wanted to discuss a bit future things which is basically we would like to have it better integrated in K develop which Well should have her soon if we find somebody to do it. If you're interested, please say hi we would like to see if How hard it would be to have plasma on those systems as well. We have two approaches to it none of them work, but we're hopeful they might someday and You have maybe half a minute for a question What is operating system? Well, Katie is not software Katie is a community, right? Plasma though is usually bundled on with the system. It's something you might Still get with your distribution, but that's not necessarily what will happen in 20 years, right?


We praise Allah as well for bringing the hearts together on this day of Jumu'ah, this day of gathering. Just as he brought our hearts together when we were brought out on the day of the Covenant, on that day when we were brought out from the loins of our father Adam, alaihi salam, Allah asked us, Allāhi stu bi rabbikum. How may not your Lord? And we testified, we all witnessed and testified. We said, balā, indeed you are. And so this gathering, this Jumu'ah is a reminder of that first gathering. And a warning of the upcoming gathering on the day of judgment, Yawm al-Qiyāmah, when we will be questioned whether we remain faithful to that original Covenant. We know then that this life is just a journey between that day of the Covenant and the day of resurrection. And in this life there are moments and opportunities when the dunya, this life that we're in starts to fade away when paradise becomes more apparent. There are opportunities that Allah gives us in this life when the dunya fades into the background and when Allah's nearness comes into the foreground. And we know that Ramadan is one of these moments. Most people are able to recognize the blessings of Ramadan. This is something that most of us experience. The expansion in one's capacity to stand for Ta'rawi and for Tahajjud, the expansion in one's ability to recite Quran, suddenly we're brdu khatims of Quran left and right. And this expansion, these blessings are so momentous that it's said that to set off the early generations of this religion, in the six months leading up to Ramadan, they would make dua that they would be able to reach Ramadan. And in the six months after Ramadan, they would make dua that their Ramadan was accepted, their fasting and their standing during that time. Yet, what is often not appreciated or overlooked about Ramadan is that this expansion in one's risk, one's spiritual risk is accompanied by a restriction, is accompanied by a fasting. Slam or somim in Arabic literally means al-qaf, it's to restrain yourself, it's restraint. And so think about this dichotomy. Allah places expansion in matters of the Akhira exactly where he places restrictions in matters of the dunya. In other words, an increase in the spiritual and metaphysical can only come when the material and the physical is lessened. This wisdom in Allah's creation is not only something theoretical and nice to know, but it has practical implications for our fasting as well. Because those who don't keep this dichotomy in mind between the material and the spiritual may inadvertently lose the benefits of Ramadan. In a famous hadith, the Prophet, sallallahu alayhi sallam says, Many are those who fast yet gain nothing from it except hunger and thirst. Many are those who fast yet gain nothing from it but hunger and thirst. Now why would this be the case? This is because when Allah tells us to fast, he's not just telling us to delay our meal times until after Maghrib. Nor is he telling us that the purpose of fasting is just to get hungry or thirsty like we would do if we skip breakfast or lunch any other time during the year. No, the purpose of fasting rather is to weaken the nafs, to weaken the physical appetites, to weaken our physical capacities. In other words, to diminish our reliance on the material world. That is, it's to wean the lower self off of its halal appetites. And that is so that the body may be freed and the soul may be freed, the Ruh may be freed to engage upwards, engage with Allah, the Sublime, the Almighty. And it's in accordance with this that the masters of the heart, the scholars of the heart outline four conditions, four restrictions if you will. For a fast to be acceptable, not acceptable in the sense of gaining reward, but acceptable in the sense of achieving the very purpose of the fast in the first place. The first of these conditions is to lower the gaze from what is unlawful.  And so whoever foregoes a glance that he wishes, a glance towards something he wishes to gaze at. If someone foregoes that, Allah grants him a faith, the sweetness of which he tastes in his heart. And today this is especially applicable to screens and to social media and to mainstream media. Because how often do you find yourself watching something or scrolling past something? And what you're scrolling past or what you're watching is something that depicts gluttony or something that depicts lust. These very things, seeing these things excites the very appetites that the fast is meant to diminish. Fasting is meant to a customer person to be satisfied with less, with needing less. The screens, as we all know, are designed to get us hooked, to want more and more. And so they are opposites, this is a dichotomy. And so this is the first condition that the scholars mention to restrict the eyes. The second condition is to restrict the tongue from its endless talking and blabber, to restrict it from lying, from backbiting, from gossip, and from arguing. The Prophet, sallallahu alayhi wa sallam, said, fasting is but a shield. Fasting is but a shield. If one of you fasts, be not foul, be not hot-headed. If a person fights with him or insults him, let him say, Inni saw him. I am fasting, I am fasting. And in another hadith, the Prophet, sallallahu alayhi wa sallam, also says, whoever does not leave qoluzur, this false testimony, and does not leave false testimony and does not leave acting in accordance with false testimony, Allah has no need for him to leave his food and his drink. So what then is the purpose of leaving our food and drink if we are still engaged with these sins of the tongue? What fasting demands is silence. What fasting demands is a tongue to restrain itself, to be fasting itself. And even the physical fasting from food encourages this because a lack of nutrition throughout the day means that a person doesn't have energy to carry a conversation with people. What it does mean though is that this lack of conversation with people frees the tongue to have a conversation with Allah, that is, through doing dhikr of Allah, through reciting the Quran. And this is the purpose for which the tongue was created. The third condition that scholars mention is to restrict the ear from listening to anything that the tongue is forbidden to speak. We know that remaining silent in the presence of backbiting if someone is backbiting in front of us, that this is forbidden. Allah says in the Quran, Sit not with them with these people who are backbiting until they delve into another conversation. If you would do so, then you would have been like them, meaning you have been sinful like them just as they are sinning by backbiting themselves.   They are always listening to lies. They are always devouring the unlawful. And so fasting includes this as well, it's the fasting of the ears. Just as the tongue is meant to recite Quran, just as it is created for the purpose of reciting Quran and for dhikr, the ear is meant to listen to these things as well. But if the ear is clogged up, if the ear can't actually hear, because it's listening to so much gossip, to so much arguing, to oftentimes to music which contains these depictions and descriptions of the unlawful, then how will there be any room remaining to listen to the Quran with heart when we read it ourselves or when we listen to it in Taraweeh? And so this is the third condition, to restrict the ears. And the fourth condition is to restrict the stomach. That is not only from eating during the fast, but also when breaking the fast. In short, we are asked not to feast or indulge ourselves during Iftar and during Suhur. In another hadith, or as we mentioned earlier, the hadith, Many are those who fast yet gain nothing except hunger and thirst. Just think about how foolish this thing is. If you were to, this is like someone who works night and day for a month to receive their paycheck. And as they're exiting the office, they throw their paycheck into the dumpster. What was the point of that work? What was the point of that month? And if you realize that that month, that work was pointless, then what about this month, this month of work, this month of fasting? What is the point if you end up gorging yourself on a 10-course meal during Iftar, filling yourself to the point that you're going to pass out?  The Prophet, peace be upon him, says, and he says this about eating halal food. He says, There is no vessel worse than the stomach that the child of Adam fills. And so we are asked to restrict the stomach, restrict the stomach even during Iftar, to the point that you should feel more hunger and thirst during the fast. Then you normally would. Because this is the exact medicine that is prescribed for us to gain spiritual strength.


So today's lecture, let me end quickly the example I was telling you about the projective plane. And then to be honest, I was dreaming of telling you something about differentiable manifolds and differential geometry, but being the last lecture, it would become just a mess. So today's lecture will be just closing this example, having a quick discussion. And then I would prefer to discuss with you exercises and problems about the course, the whole course. So it will be a special strange lecture. I warn you right from the beginning. Well the end of the story about the projective plane was this. We defined one differentiable structure even though we presented it in two different ways on P to R. Essentially one was coming naturally looking at it as R3 minus the origin quotient by the equivalence relation. And the other was thinking of it as the quotient of S2, modulo, the antipodal map. This induced naturally two seemingly different differentiable structures here, but it's easy to check that they are actually the filmorphic. You can write one change of charts in the first differentiable structure in terms of a chart of the second structure and you see this is a differentiable map and vice versa. That means that the differentiable structures are really the same. And then I mentioned to you the problem that whether a differentiable manifold comes actually as a sub-manifold of Euclidean space and this object here fails to be a sub-manifold of R3. We can call it a surface in the sense it's a two-dimensional manifold but it's impossible to embed it inside R3. So the idea is just to, I mean, if we want to have some simple model of this, on the other hand I mentioned to you Whitney's theorem, I didn't tell you the name last time.  And in fact in general if you take an n-dimensional manifold, compact n-dimensional manifold, this will always embed inside R2n plus 1. So this is kind of the estimate we have for the dimension of the Euclidean space where you can see n-dimensional manifolds. So applied to surfaces this would give R5 and I mean it's usually not easy to prove that something cannot be embedded in one, it's not a simple problem. On the other hand, let me show you why P2R goes inside R4, which is actually less than the general dimension. Well, the way to do it is to construct a map which I call phi. Look at phi from R3 to R4 given by these functions.  You see, basically you are taking a subset of polynomials of degree 2 in the variables, in the variables x, y, z. And remember that P2R actually so. You see, if you think of P2R as the quotient of R3 minus the origin, quotient, the equivalence relation given you by multiplication of a non-zero number, this map doesn't pass to the quotient, because the point, the equivalence class x, y, z would be, I mean, a point x, y, z will be equivalent to lambda x, lambda y, lambda z. And the image of lambda x, lambda y, lambda z is not, it's completely different, but it will be lambda squared. But it's not the same point. Here we don't have an equivalence relationship. So this is not a map which passes automatically to the quotient. The way to see, the way to correct this problem is to look at this map on S2. Because what is the difference, of course, which are the, which is the equivalence relationship on S2, which gives you the projective plane, is the antipodal map. So P minus P are the only two representative of a given equivalence class. But P and minus P will go to the same point. Because if I take minus x, minus y, minus x, z, these are quadratic polynomials and they kill the minus. So this map goes, when I see, when I restrict it to, so on S2, phi passes to the quotient, the quotient giving P2R. So in fact, the quotient map, I can define the quotient map, let me call it theta if you want. So there is a map from P2R into R4. This will be formally the quotient map of phi. So now the claim is that this is actually an embedding. Well, what do you have to check? You have to check many things. Most of them are completely, in fact, almost all are completely trivial in the sense that, for example, is this map injective, first of all? Well, that's simple. Suppose that there are x, y, z, and x tilde, y tilde, z tilde going into the same thing. And then you have to argue that the two points differ by a minus. The only possibility is that they differ by a minus, for example. Then, well, is it continuous? Well, this map is certainly continuous because it's a differentiable map of an R3, a stric to 2S2 quotient. So with a quotient topology, this is by definition a continuous map. Then what else? Well, then, is it differentiable? Well, to check if a map on a differentiable manifold is differentiable, you have to write it in terms of a chart. But in terms of a chart, which chart, for example, now we have two nice sets of charts in some sense because we have defined two possible ways to describe the differentiable structure of this object. In terms of, remember, in terms of the maps of the charts given by from the disk to the sphere, okay, when we describe it in the second way, we define, for example, the map F3 plus, which was taking the disk into the upper sphere with respect to the coordinate z. So that means it's taking the point x, y, and giving you x, y square root of 1 minus x square minus y square, with a plus here, otherwise it was a minus here, okay? So this was one, for example, one possible chart. From the disk, no, x, y move in the disk x square plus y square less than 1 and gives me this point here, okay, on S2, on a part of S2 where each point has only one equivalent point. It's equivalent only to itself, okay? So I see this map with values in P to R, okay? So now how do I check that the map is differentiable? Well, I write phi composed F3 plus, for example. Of course, I should do it for any i and plus and minus, but they are all the same, okay? Up to changing of names and putting a minus somewhere, okay? So what does this become? This becomes the map, right? This becomes the map x square because now you take this point and you have to compose with this map here. So this becomes x square minus y squared, x, y, and then let me just say, every time I see a z, I call this function here d, okay, just to give it a short name. And this becomes xd, yd, okay? So this is the composition. So there are questions here. Of course, this is another way to check that this is a differentiable map, okay? Because now I wrote it in terms of a chart. So I wrote the map phi in terms of the chart, and I see it's still a differentiable map. Now the problem is, for example, is the differential injective, remember, I'm trying to prove that this is an embedding. So I need to prove that the differential is injective, okay? Is the differential injective? Well, the differential, I need to compute d of phi composed of 3 plus at any given point, okay? I leave the point free because it has to be true at every point. So, well, but this object here now is a map from R2 to R4, okay? Because now this goes from the disk to R4. So the tangent space to the disk is R2, and this goes to the tangent space to R4, which is R4. So this is represented by what? Is represented by the Jacobian matrix, by the matrix, which I construct by taking partial derivatives in the rows, for example, no? So if I take, so on the first row, I put the partial derivative with respect to x. So this becomes 2x, y, y, d plus x dx, y dx. Remember, d is a function of x and y. So I leave, I indicate by the partial derivative as usual. And then I take, in the second row, I put the derivatives with respect to y. So this becomes minus 2y, x, xdy, d plus y dy, okay? And now the question is, has this matrix rank 2 at every point? Because now this is of dimension 2, and the requirement of being in dimension is that the differential is injective. So it has to have maximal rank, and the maximal rank here is 2, okay? Is it 2 or not? Well, you see, it's enough to look here. This minor, it's twice x squared plus y squared, okay? So that's okay away from 0, 0, 0, okay? 0, 0 is inside our domain, because we are taking x and y in the disk, okay? So this would be a nice minor, except from 0, and then you look for another one, okay? And then what else? Well, you need to, what would need really to prove, so it's clearly, so it's a differentiable map. It's an immersion, it's one to one. It's continuous. Well, you would need to prove that it's an homomorphism onto its image, okay? I skipped this. It's purely topological problem, okay? That's the only bit missing. In any case, this was just an exercise in this, okay? Now as I said, P2R is compact. Okay, yeah. If you want, you can use even, you can bypass the problem in this one, okay? Now, as I said, I was dreaming of telling you more about the beginning of the story on the differentiable manifold, but I hope you realize that all the things we have done for surfaces have been already done in a way to prepare you to jump to this more general setting, okay? Now there are of course points where things get delicate, which we didn't really see in the theory of surfaces, because using the fact that there was an R3 around, we could bypass the problem going out from the surface into R3. For example, orientability, I mean a key point, a key thing that you want to do on a manifold is of course integration, okay? You would like to do two things, differentiated integrative. You want to have a reasonable calculus on a manifold. For example, to integrate, we need orientation. Now orientation in R3, for surfaces, in R3 was easy to bypass by asking that there was a globally defined normal vector field, okay? This was a way to say that something was oriented. But of course, if your manifold is an abstract manifold, it doesn't live anywhere, it doesn't make any sense to speak about the normal vector. So how can you solve the problem of this? Well, that's for example another point where you really think of what you need and you put it as a definition, but then it's interesting to realize that this is exactly what you had in mind for surfaces in R3. Namely, for example, the problem of orientation is that is to ask that your manifold, of course it's always covered by charts, this by definition. And you know that when you overlap, there is a differentiable function, okay, in the other picture from here, the usual picture. Basically, you ask that this function here has positive Jacobian, okay? So this is another way to say that these two charts are oriented in the same way, okay? And in fact, this corresponds precisely to the fact that remember when you did xu cross xv on one side, so if you had two charts and y, you tilt the wedge, you had the two things. Well, of course, the fact that these two things match corresponds exactly to the fact that the Jacobian of the transformation giving you, for example, the y in terms of dx is positive. Because here the only thing that can happen if you think of R3 around this object is that your normal vector, when you pass to the other chart, becomes minus it. So being positive, the Jacobian is exactly the fact that the standard normal vector on one chart goes to the standard normal vector in the other chart. Well, of course, normalized, okay? I'm not asking to be equal to 1 because actually here, there is divided by the norm. So the only thing I care is if it's positive or negative, okay? You see, this, for example, it's another point where you have to stop and think. If you give this other definition, so an orientable manifold will be a differentiable manifold which is covered by charts in such a way that every time two charts overlap, the Jacobian is positive. In the case of surfaces, this boils down to the old definition. But this condition is exactly what you need to define an integral, okay? Because the Jacobian of the transformation is exactly what appears in the change of variable formula for integrals, okay? Up to the absolute value, that's the kind of the problem. So when you want to give the notion of the integral of something, in principle, it would be defined up to a plus minus, okay? Because of this problem. If you know that your manifold is orientable, this problem disappears, okay? I'm not trying to, but this is another moment where, so integration of what, but actually the other problem is what can you integrate on a manifold? And there, that was, I was hoping to, I mean, to describe you quickly but it's impossible. So differential forms, whatever, if you know it already, that's kind of the natural object to integrate on a manifold, okay? Now, now what else? Well, so as I said, this is just the beginning of the story. Basically, we did the first maybe 15 pages of any book in introduction to differential geometry. Then how things go on? Well, you see that even just by having the definition, you have some beautiful questions that you can think of. So for example, this notion of differentiable structure, the differentiable structure induces a topological structure. You haven't stopped me, but clearly it's a question that's behind our mind. Is it possible for two differentiable structures to induce the same topology but they are not defiomorphic? Meaning, the association from differentiable structure to topological structure. Is it injective in some sense? Notice that the two equivalence relationships are different because on differentiable structure you identify defiomorphic things. On topological spaces you identify homeomorphic things. So basically the question is, is it possible that two homeomorphic manifolds are not defiomorphic? So this is already a beautiful piece of mathematics. You don't need. The point is, okay, the question is simple, it's beautiful, it's fundamental, but the answer is highly non-trivial, okay? It turns out that this kind of, because you see, for example, for surfaces you can prove that this is not possible. Two surfaces are homeomorphic if they have a differentiable manifold. Two surfaces are homeomorphic if and only if they are defiomorphic. So you need to go in higher dimension to understand whether this is possible or not. And the answer is actually yes. This was a great surprise. Even for compact manifolds and in fact even for very simple topological manifolds. So Milner, John Milner, which is probably a name you find, you found that in many places in mathematics, John Milner, when he was horribly young, proved that even among spheres you can put. So if you take SN, if you start from seven, these topological spaces have differentiable structure which are not defiomorphic. This was really a surprise. And these are called exotic structures of course. You will find even books on the description of this. Of course, if the sphere has it, you can produce it on many other compact manifolds. But then there was a second, so this was about 1954, if I remember well. But then there was a second part of the story where you say, okay, maybe you have to go in very high dimension and you have to take, what about Euclidean spaces? So if this phenomenon happens, you can ask, does it happen on the simplest possible manifold? This is the simplest compact manifold. What about Rn? So you fix the Euclidean topology, well no, you don't fix it, you are there topological structure which admit, which come from differentiable structures. And as topological spaces, they are homeomorphic, but as differentiable manifolds they are not defiomorphic. And this was even a bigger surprise because the answer is again yes, but in this case it's a very strange yes because exotic structures exist on Rn only in dimension 4. So only for n equal to 4, these structures exist. And in fact in dimension 4 there are infinitely many, more than countable many such structures. And this was a beautiful theorem by Donaldson and Friedman. This was about 1984. So even R4, which is one of the most familiar thing you can think of, produces this incredible phenomenon and there is still a lot of research to try to understand what is the geometry of these manifolds. They are genuinely manifolds. For example, so this is one way, even just with the things that we have just defined it's enough to produce. But then just to indicate, probably many of you have already studied something that is close to what I'm saying. So the game, what is the game in this, I mean, is to try to associate, so to classify manifolds under special assumption. Like we did, for example, curvature, I mean, how curvature condition on a surface determines which surface it is. So for example, a surface of constant Gauss curvature, a compact surface of constant Gauss curvature in R3 is the sphere, for example. Or we did the classification even when it's not positive. So this type of theorem. Well, in differential geometry, what you try to do, you see, the differentiable manifolds are too flexible. I mean, you cannot really, I mean, they are very difficult to classify in the sense that there are too many differentiable maps. Differentiable maps are really flexible. You can stretch and push a manifold wherever you want almost. So the nice, the beautiful and difficult thing is to associate to a manifold some numbers. Think to the case of a surface, when we associate to a surface, a compact surface, the Euler characteristic, for example. This is the prototype game, okay? You have a geometrical object and you associate a number. And then you ask, is it true that this number classifies? So is it true that if you have, I mean, this number must be invariant, for example, in this case for homeomorphism, okay? So it's a topological number. And then you dream of a theorem that tells you if two surfaces have the same number, they are homeomorphic. That this association is somehow injective, okay? And in the case of surface, compact orientable surfaces or compact non-orientable surfaces, remember, but you cannot mix the two together, this is true, okay? Two compact orientable surfaces with the same Euler characteristic are, in fact, the same. So if you want to mimic this, so this is beautiful, I mean, this is great, okay? In terms, I mean, for a geometer, this is the best you can hope, okay? At least as a first step. Is there a similar game you can produce here? Well, there are many of these type of games, okay? And for example, I mean, you know some algebraic topology, you can associate groups instead of a number, no? You can, for example, you can associate to a topological space its first fundamental group, okay?  Groups, corresponding groups, then they are the same, okay? Well, in general, this is not true, of course.  But if you believe the theorem that a surface with a given Euler characteristic is given by a sphere attaching handles, no, as more or less, I mean, the Euler characteristic is measuring by 2 minus 2g, g is the number of holes of a surface. Then it becomes also true that the first fundamental group, I mean, two surfaces with two different number of holes will have two different fundamental groups. But for example, the sphere will have fundamental group trivial, the group 1, okay? Every loop on the sphere on S2 collapses continuously to a point, okay? If you take a torus, one hole, what will be the first fundamental group of this? You have two independent generators, and there are no relationship between them, okay? There are no abstract relationship between these two loops, okay? So it's a z plus z. So you see, in this case, so for closed compact surfaces, again, for example, orientable by, you can rephrase the first fundamental group, the term is the surface, okay? But again, in general, is it true if you go to higher dimension, for example? Well, for example, even in dimension 3, this was the content of one of the most famous problem in mathematics, what it was called the Poincare conjecture. If you take a compact three-dimensional manifold whose first fundamental group is trivial, so restrict yourself to the simplest case, so no holes. Suppose you have a manifold, compact three-manifold whose first fundamental group is trivial. What can you say? Well, Poincare asked, in fact, he asked it in any dimension, okay? Is it, but then if you go in dimension higher, you have to start putting also the higher or more to P-type trivial, okay? So in dimension 3, there is only one left, which is pi 1. Otherwise, I think you have seen in the course of algebraic topology that there are higher homotopy, okay? In any case, so in any dimension, you would ask, if something has trivial homotopy, okay, is it a sphere? This was the content of the Poincare conjecture, okay? And this turns out to be true, okay? And it was proved, it's a beautiful, I mean, this is, in my opinion, it's probably the most beautiful theorem in geometry that I know, okay? Also because of the way it was proved, because it's very surprising. So at the end of the 60s, it was proved for any dimension greater than or equal to 5. And in fact, now we all kind of believe that this is easy. It was one of the triumphs of Morse theory, okay? With some Morse theory, with some delicate, but I mean, not Morse theory, you can prove the Poincare conjecture in any dimension greater than or equal to 5. So in some sense, the problem in dimension 1,500,000 is much, much easier than in dimension 4, where you can think that maybe with some, in fact, it's much, much, much easier than in dimension 3, where you can even try to draw a picture, okay?  So in the, by Stephen's Mail, Stephen's Mail proved, I mean, of course, all these people, each of these theorems is a Fields Medal, no? I mean, these are kind of milestones in geometry, I mean, these are not theorems like anything else. Now Stephen's Mail got the Fields Medal for this theorem. And then in dimension 4, with the very Hadoch four-dimensional proof, Friedman got the Field Medal for the solution for S4, and then it was left three dimensions. Strangely enough, this was resisting every attack, and this was proved in 2,000, well, unfortunately, there is not really a date because probably you know it's a strange story. So these were between 2003 and 2006 by Grisha Perelman, okay? Using partial differential equations, essentially, technique, okay? This was the really the masterpiece of mixing analysis and geometry and topology all together, I mean, even though what I said would be true also for this part of the story. When you get to this level of questions, mathematics is one.  And that's why it's also beautiful, okay? Now the story was slightly complicated because of course Perelman was, well, he had his ideas about the way to communicate science. So he posted on the web two preprints, well, two papers. But he didn't want to publish them, which is the usual way the mathematical community decides whether a paper is correct or not. Or I mean, he just put on the web two papers and he disappeared, okay? And in fact, in those papers, there is not even the statement. He was not even claiming the Poincare conjecture. He was proving something that everybody knew. It was very closely related to the Poincare conjecture. But he was not even bothered by saying then the Poincare conjecture is proved. So the mathematical community was kind of shaped to say, well, now what do we do? I mean, first, are these theorems correct or not? And so many teams of mathematicians all over the world started working to check whether what he did was correct. And then how to use these theorems, that was kind of the easiest part in some sense, but I mean how to use these theorems to finish the proof of the Poincare conjecture and that was done, okay? So in 2006, there was the International Mathematical Congress, which is the moment where it happens every four years. So it was the moment where we give, I mean, the community gives Fields Medal to, and it was proposed that Perriman would have gotten a Fields Medal for that. But he refused, okay? He didn't want to get prizes for that. He was kind of, he did it for his own, I mean, for the advancement of science, okay, and not for getting a prize. Which is, I mean, if you want, it's kind of strange, but it's noble. I mean, we have to admire him also for this, because actually attached to the Poincare conjecture, there was a $1 million prize.  Well, especially because he was not the son of a gas tycoon in Russia.  Anyway, okay, so for example, you see, but going back to now, end of gossip, going back, I hope I'm trying to, I'm communicating you a strategy, I mean, classifying manifolds. How you try to classify manifolds? You try to associate to manifolds numbers or groups or vector spaces or something where you can think that, I mean, the classification of the other object is much easier. For example, if it's a finite group, you have the classification of finite groups. If it's a vector space of finite dimension, in fact, it's r to that dimension, okay? So you don't, again, have to classify anything. So in that case, of course, the only interesting thing would be the dimension. And then you reverse, so there is a first part where you construct this object. So there is this association. And then there is the problem of, how delicate the association is. I mean, two manifolds with the same objects associated, are the same or not, okay? And this game goes on since it's already a century that this goes on. And it will go on probably for another century, if not forever, okay? Because, of course, we have not identified numbers. So such delicate numbers to tell us, well, if these numbers are the same than the manifolds is one of these, okay? There are many of these attempts and every time we, so for every time we have a list, we separate parts of the list into smaller pieces and so on. But I mean, this is part of current research in this general philosophy. The first game you would do in differential geometries to associate what is called a homology and homology, okay? But that's another, that's the story, for example, there are homology. It's a way to associate to a manifold a vector space, I mean, a vector space and okay. But there is definitely no way that in 40 minutes I can give you an idea on this. Well, okay, so in some sense I think the course is over and the second part of this lecture I would like to discuss with you problems, okay? Exercises, okay, so now let's go back to some of the exercises I proposed to you during the course. So I'm asking you, one of your colleagues to help me doing it, okay? Now, the exercise asks basically to classify isometries of the helicoid, okay? So, let's see how we do it. So the first question is to prove that K is, the Gauss curvature is given by this formula, okay? How do we do it? Okay, so just to save a bit of time. Now there are basically two possibilities.  And you take this function and you apply it in the formula, giving the Gauss curvature which comes up from the Teorema egregium and assume, okay, and you get this function, one minus, right? Okay, so now since now we reduced it to just computing a derivative, this is the output. Okay, so let's move to the second question, okay? The second question asks if you take a point, if you have an isometry and if you take a point and it's image point, they lie at the same distance from the z-axis, okay? So how do we argue for this? So we can write any point, sorry, I repeat because of the microphone. Okay, we can write any point on this. We write any point as the image of- So image also can be written in the same way, at some point, some YouTuber, depending on this year. Okay, there will be another choice of parameters for which this is true. So let's say I use this formula to give the z-axis, we will give the z-axis also. Okay. But from here, we know that k of p is the z-axis k of p. So by the theorem I grade you, we know that these two points must have the same gas curvature. So this was a direct application of the theorem I grade you, essentially, okay? Now, in fact, because of course the exercise would be write down all the isometries, okay? So all these steps are to get to the final point. So now let's prove another intermediate step. So f, in fact, so what the exercise suggests is that f acts as a translation on the z-axis, okay? Possibly after a reflection. So let's see how we can argue about this. Your u is very, very similar to a y. Okay, well, call it t. This we can see is exactly when we put v to zero x of u zero. Okay, these are the points of the form. Okay. So if we restrict f on this one, so we have a map. Because any element here is uniquely determined by this t. Okay, basically we are looking. This is kind of the restriction of the isometry on the z-axis. Well, I said it a bit brutally because we still don't know that the point on the z-axis stays on the z-axis. So, but let's see. From this one, so here, I know that for any element, for all t, the distance between x. So in fact, this answers my problem because that means that if I take a point on the z-axis, this will stay on the z-axis. So this map g is in fact, I can write the map g as a map from r to r if you want, okay? Okay. Why is the map different from r to r? Yes, for some function k. Okay, what you have is g prime. If I use this formula, this is f on point x with respect to u, zero. And this using this one, this will give me exactly zero. Okay. Now, f is an isometry. So you present the first fundamental form. So what I have a negative g is exactly 0. And this will give me 0, I have this is 1. So this will give me 1, this is z, and this one, which is half the value of the prime point. Okay. So this will give me, I'm using continuity of t. So I have a negative key of t, k prime of t is exactly 1 of t, k prime of t, now this one, we obtain that. So that means k of t is equal to, well, okay, okay, let's not, I mean, that means that k of t is equal to plus or minus t plus c, plus c, okay, plus some constant. And that's exactly what you had to prove, because the minus sign here is exactly the reflection in the z-axis, on the z equal to zero axis. Okay. So we also achieved 0.3 in our staircase. Now, 0.4 maps a ruling to a ruling. Well, actually I don't know if we use the word ruling in the same way during my lectures. Of course, this means just, so this is a ruled surface. Okay. So that means just it maps one of these lines to possibly another of these lines, okay? Maybe we can erase this. If the space is enough, yeah. If you can speak a bit louder, I think. From this expression, you can write x of u v cos of u. So if I do this as alpha of u, any ruling on this one, x of u v of r, so mark is the image of this. So the written in mathematical form, what you are asking is, is it true that f of l u is equal to l u tilde, for some u tilde? Okay. Now that we have translated the problem into mathematical terms, you fix you, yeah. Mm-hmm. Yes? f of v is exactly, now here, u is fixed, v which is moving. So I think that any element can be written in this form. So this can be written as sum of this one, sum h, perhaps h, if I put this one inside here. Now u is fixed, the v which is moving, so this is, so if I use this inequality, I have this prime always equal to this one, because here u is fixed, this v, if I use this equality, this will give me, so from here I have that the number of this prime, this prime is exactly the number of f and this using the fact that f is an isometry, this is exactly f tilde, this is exactly that. Sorry, may I interrupt you, because it seems to me we are taking a root which is too long. We know already that the map f preserves distances from the z-axis. So you already know the function g2. So what you have is g2, the only thing you have is g2. So it's plus or minus. Wouldn't it be easier to use this information right from the beginning? You can, for example, assume it's plus. Of course, the difference between plus and minus, the only problem is when it's zero. So if you assume it's plus and you are not on v equal to zero, you can argue. Then of course the same proof will work when this is minus v and you are away from zero, but then the zero case you have already dealt on a side. So somehow you don't have to worry about the fact that you change. So in some sense I think if you write, without loss of generality you can assume g2 is the identity. So here you have data of this point of this because this is exactly the square. Now here so we can also say this is exactly f composed of x and z1. Now here we fix u. So we have also, just also, this is exactly g1, which is y. This is 1. That is equal to 1. G prime is equal to zero. Call it d, I don't know. Just to avoid confusion with this. d is constants. So if you put it back inside, what you have is... Well, okay, now it's okay. Because you are just changing u into another thing, which is constant. So it's done. And now let's put everything together and give the classification of isometries. So what have we learned out of these? This one, if we suppose that u is fixed, it will take two... Sorry, this is a u. Okay. We learned that this one from the second question, this take you wanted to, another one. So if you suppose that this was fixed, this should be normally half of some u, but depending on this u, this is fixed. I think it would be more clear if you would call it b of u. I mean, f, by the previous question, you know that u goes into something like this. But here what is not written here is that, of course, everything could possibly depend on u. Okay, so let's call it b of u. Either this one or minus. So if you suppose that is v. So we have v plus w. And we know that... I want to get this formula explicitly. So I have here that if v is equal to zero, I have zero, zero is exactly alpha. And we know this element belongs to the axis. So this is the translation. And this is the z to zero, zero, u for some constants. That's what we proved in part three or two. No, three. No? If you put inside, it's done, because now what's written there is just the x of u plus cv. Okay. So were there alternative solutions? Anybody found some... It's a symmetry. If u is minus u... If u maps to minus u... Yes. But no, no, it's not reflection, I think. It's a rotation in the x-axis by 180 degree. Because u is also in other sides. It's not reflection. Because u is also in... So I think it's a rotation. It's a composition of the two, no? Because u goes to minus u, it's the reflection in the z-axis. But at the same time in the x-y plane, you are doing something, no? No, I think only u to minus u, it's all the rotation. The rotation where then? The rotation in the x-axis. So come to the blackboard, because I mean in the x-axis, I don't know what it means. We wrote the thing in the x-axis, v to minus v. Okay, sorry. What were you arguing?  But if it's the reflection, this must not say... If this is a reflection, then the minus must not be here. Right? So I'll give the resist rotation in the x-axis. Sorry, I don't know what it is. A rotation in the x-axis. I mean... No, but suppose just... Let's draw it... You have R3. So first I don't know why you are keeping... At which point this should be relevant in this exercise? Because here... Okay, so you want to say that this is up to something. I mean, so there are these family and another family. Okay, but now what you wrote? So geometrically it takes this point where... I mean, so this is x, y and z. Right, so it's a composition of... It's only a rotation in... Well, so the rotation means it's something going like this? Yes. So it's a rotation where the axis depends on the point. The axis is here. So if this is the cylinder, then the rotation is so... Okay, we can discuss it later. I mean, the point... In any case, so the point that you are making is where is the reflection gone in this argument? Yes. Okay, now forget... I think you are a bit confused, but it's okay. What is the geometric meaning? But what you are raising, you are raising the question whether we should add here a plus or minus? Okay. And also v can be a minus. Okay, so let's see. What do you think? Yeah. So in fact, I think you have to allow this freedom. The only point is that probably you are writing twice... I mean, it can happen that you are writing twice the same thing. Okay, depending on the way you write. But actually, even in my solution... No, no, but actually, no. The point is that rotations... I mean, reflection should disappear. I mean, reflection along the z-axis are not there. So in principle, you leave it here, but then you have to argue that they disappear. Okay? Maybe it's still on the blackboard. It's here the point. B of u is not plus or minus u plus c. It's plus u plus c. So this thing here tells you that actually reflections were not there. Okay. So at the end, these are the only isometries. Well, okay. I mean, of course, I have prepared six exercises and we have done one, but I hope... Well, we can also have a special lecture on exercises if you want. Do you think it would be useful? So we will plan it, okay?