Hi, my friends call me Lyme and this is what you have to say about expectations. Abandoned New Zealand, why did you abandon New Zealand? It says that I always make easy goals I know I can reach so no one gets disappointed. Now setting up attainable goals is one thing but making things easy for yourself is another and that might be a problem because the more you challenge yourself the faster you grow and safe content might not be as entertaining as it could be so I guess balance here is the key. This kind of reminds me about people telling me never to get my hopes up because I might get disappointed. I kind of refuse to harden my heart to never experience one of the greatest feelings that life has to offer a hope just because I'm afraid of getting disappointed. TORORUS TORORUS says and I totally agree with Lyme's point you must not always set high expectations because you might not achieve what you wanted and feel bad for it so you must think realistically and always keep trying for an improvement. Yes exactly setting realistic goals is the key you keep improving at the same time as you reach the expectations of your audience. Dr. Dunk says just make the best quality video you can in a certain time. Yes you should always try your best I mean of course you should however the point I was trying to make is that there are situations where doing your best in any given situation risks creating too much contrast in quality between the different contents of your channel and it's that contrast that makes your audience expect things that you then don't deliver and that's when you shoot yourself in the foot. RECEPTION says hey guys I need some help with a logo and banner my current one doesn't feel as professional as some others I see. Don't do professional banners because everybody else is doing them do professional banners because you offer professional videos otherwise you're setting the wrong expectations. I am though totally open to discussing how much banners actually contribute to setting expectations I mean maybe viewers today are just used to banners not reflecting the quality of the videos but I still stand by that consistency is the better option of the two and I mean not professional doesn't have to be equal to ugly some things can be very beautiful and feel homemade at the same time so don't worry about professional worry about reflecting the content of your videos. So I hope that cleared things up and in case you're covering my face right now with all the stuff I'm gonna do this thank you so much for watching bye


Oh jeez this is fuzz Okay I'm trying to make him the studio cat. I Don't know if he likes it. So I got him some tuna. There's your tuna fuzz. Okay, let's go day one I'm just gonna start If I don't know where to start sometimes. I'll just You do like the stuff that's I'm painting edges black Just painting edges black Just get you started and then it moves stuff forward, right? See Okay, we'll have to work on the studio cat thing fuzz Teddy here. Come here. Ted Ted Ted Sorry man pet management here. Teddy. Can you grab that please? That fuzz wants to go upstairs. I Don't know Okay, well that Rebecca solved that every morning a Beef for breakfast and ginger tea. Hmm. I wish I could play music Well, it was live because usually right now actually I was listening to dub reggae DUB dub is this working guys the angle and the audio and stuff I'm just gonna stop talking You couldn't open this All right one thing that we always do is we don't put the lids back on our paint and then you got to fight Marino It's always handy the big tubs of black and big tubs of white. I have a faster way to do edges actually I'll straight the faster way calligraphy ink some save your old containers When your kids grow up, I'm not gonna eat on camera There's lots of water for drinking. I like this one So all the projects that I'm working on are things that I started last year when I did the hundred days of art I wasn't making any art. I was making zero art Then I did a hundred days of art now. I have so many projects on the go and I'm excited to make art. Okay, so Hit the towel Indian ink Look at that. Hey, I should just always do it this way And well painting is getting worked on they actually get dirty again and not great See Rebecca just said she wanted yellow in our house somewhere So I just put a bunch of yellow down then I put it there and I just love it Yellow a bit orange and some gold so I'm gonna I'm just doing this to the edge and Then I'll do some touch-ups and then I'm gonna resin it I'm gonna resin it and it's gonna look amazing and look at that Okay, and then this one I want to do edges too See I done that just I did edges a while ago, but then they get covered again as you work on it. So just a quick wipe Just makes everything tidy I hope this is helpful to someone seeing my process and just what I'm doing because I Hope you realize that you just kind of make it up as you go. You just got to start make it up as you go And that's kind of the joy of getting lost in making art Just freaking do it.  I'm doing I've been one of these chicks. Oops. I just got black all over there I'm doing one of these chicks For each season. It's gonna be great. Okay Great now. I'm warmed up so this one I've been working on this piece for 20 years and I'm so close. I just don't know what to put in this circle. So this is one thing that I do I'm gonna need a brush paint brush Okay, so pallets old old placemats. These are kids toys Pieces of plexiglass At home Depot you can sometimes get free off free pieces of plexiglass put these out. I love having water bottles all around Because then you can you just get some neat things happening good next so I want to I want to put this on here. That's what I want to do Crayons work on here. Oh, yeah crayons work great I need a cat in here 76 divided by 2. Oh my gosh, well 35 times to 70 Okay There's center So this turtle is very mathematical very mathematical So I'll divide it into thirds a border here the same third third Border so there's our center line, right? So I painted a canvas black then I resented it then I sanded it and I'm gonna paint on top and Then I'm gonna resin it. I'll do so I'm just gonna eyeball weight. Okay, so there's four quarters. So Let's do the quarters like that doesn't have to be perfect there now What's the ratio from this should be halfway?  So this is where I'm gonna put my center diamond basically So here's just a few more position things So it seems like a lot of people are gonna do this hundred days of art. I had a friend Paul over for New Year's last night and he said he's gonna do it and I didn't expect that and that's freaking awesome Okay So here's the cool thing about this turtle is watch this line you got this it makes that diamond It comes it makes the legs it makes the diamond and it goes back up. Oh, I'm sorry. Can you see that? I don't know if you can or not Leg comes around makes the middle diamond leg leg makes the diamond see that shape Like will shape though. I love this kind of stuff geometry In art And I got this this book power of limits Proportions harmonies in nature art and architecture So it's all about beautiful shapes that are mathematical. So they just feel right You know There's some really beautiful shapes Like even like look at that shape You know like it just is a beautiful shape.  That's what I'm gonna do. So finds the proportions mathematically in a wing for example, and then It follows this basic rule Anyway, so that's what I'm trying to do here, right? There's a line line that comes down like this It's a circle here Makes this side of the triangle. Okay, and then the lines go to where halfway mark So that's the main idea for this painting as far as I've got so we were in Mexico three weeks ago and That's where I did the sketch because there were tons of turtles there like it's counted 23. I think was the highest so Yes, so that's where I got the idea for this and now I'm transferring it here and now I don't know what I'm gonna do I don't know what the next step is. I'll probably just start adding color to start adding color and see what happens Okay, and in Mexico I got bit by the brown recluse spider So it's almost better That's the bite.  You got to ruin it So I'm doing this kind of shape or something Okay, so extra paint like I put too much paint on there. It's really good for Putting bases on other paintings. Hey, so here perfect and over time we can sharpen up these Edges and you have more layers and different colors and Sometimes the painting will take 20 years. Sometimes the painting will take 20 minutes. I'm starting to get tired New year's party last night This uh, okay, so now I can start Shaping this foot properly, you know, I'm being more careful because I know I'm videotaping Like I wouldn't have done that The hands are kind of messed up too, and I got to fix this shoulder right here to fix it. You got to break it All right next So this piece I did this piece a long time ago, but I never really finished it and I built this black frame for it So this was when me and Rebecca were first dating. I took a picture of her walking towards me And I think it's gonna go with this piece this big one here So I want to make the match somehow so I was gonna I'm just gonna make them have some things in common So they fit nicely together Okay, what do you think pretty square? So here's the plan paper towel Indian ink Water I drink out of this skull thing Because it makes me feel strong and unafraid I'm afraid okay, so Just to let you know I started painting every day As a challenge because I was desperate. I was desperately anxious. I Was suffering All day all night suffering so Making art every day really impacted me positively very positively I feel really good most days So this is just coat one and I'll be able to look at it, you know and see if it was a good idea or not Remember you're not making art Well, you can make art for other people, but I Try really hard to not think about Who might see it? Well, I don't know if that's true So if you do this every day, you'll start to have things started you start to have projects Then it gets easier because you're not starting from nothing You you know what you're gonna do tomorrow and then you start to look forward to it So you're just every day or just like push it, you know nudging something forward a little bit nudging something forward a little bit and Then you start having thoughts about that in your mind Like what am I gonna do that piece Also, it doesn't have to be painting you can just even baking I was reading a book about someone that they started baking every day And it had a similar positive impact on their life I think if you just baked every day, how good would you feel? You can find healthy baking recipes I mean even if you like have baking a healthy baking challenge, you know, maybe you'll have You'll have a really good diet and that'll even make you feel really better And you get to bake because you love it. Just making art just Gives you your life back Okay, I'm slowing down. I don't want to do anything else Can I stop can I stop? Okay, everyone. I hope that was helpful for someone to know like how to just do it how to start and If you have any questions, let me know and if you want me to film like other stuff like organizing organizing stuff or like finishing stuff or Setting up stuff if that's helpful to see Let me know and I can just turn this camera on anytime Thank you everyone who's doing this with me Thank you. It's gonna be really really good. Happy New Year


President Bola Tinibu has signed the National Students' Loald Amendment Bill into law. The President signed the bill in a short ceremony at the State House in the presence of the President of the Senate, Senator Gotswila Pabiu, Deputy President of the Senate, Barao Gibrin, the Chief of Staff to the President, Femi Bajabia Mila, the Minister and Minister of State for Education, the Minister of State for Youth, Ayodele Olawande, Minister of Arts, Culture and Creative Economy, Hanatu Musawa, the President of the National Association of Nigerian Students, other government officials and aides to the President. Speaking afterwards, the President thanked members of the National Assembly for their expeditious handling of this bill considering the children of Nigeria and the importance of education. We have just signed the bill programming the student loan effectively. First of all, I must thank the members of the National Assembly for their expeditious handling of this bill considering the children of Nigeria that education is the tool to fight against poverty effectively. We are determined to ensure education is given the proper attention. Who is there for you to apply no matter who you are as long as you are a Nigerian citizen.


What's the most memorable vacation you've taken? Memorable vacation in Maldives. I think there's a girl that I was drowning, so I went down trying to save her, and I almost died trying to swim. I did not expect the story to go that way. I hurt girl, I hurt Maldives. Yeah, trying to swim back. Yeah, it was, I was cutting it close, but... Wow, did she make it? No, I wasn't making it. I had, you know, had them. No, no, no, did she make it? Did she make it? No, did she make it? Yeah, she made it, she made it.


You may have seen that we recently hit 80,000 subscribers on my YouTube channel last week, but additionally I just hit, what is it, one million hours of Luke Smith content consumed on YouTube That's a lot of hours and 12 million views. I remember last year we hit 6 million views And we've already doubled that now of course in the grand scheme of things That's like the amount of views the PewDiePie gets in a day But it's great to see the channel has been I guess advancing especially now that I've been putting out videos anyway in this video I want to answer some user questions. That's why I have my laptop out here So I want to answer some user questions FAQ that I actually get a lot and a lot of them are about Some of them are about Linux, but a lot of them are about me lifestyle stuff like that So anyway, I'll read I'll read the first one out first one is actually more of a Linux question Actually, maybe it's not a Linux question. I'll let you decide. So here's what this is something I actually get a lot so I wanted to do an example of this Do you think the time you've spent learning about all this stuff is actually outweighed by the time you've apparently saved? Not a critique just the thought that's a good question because most of the times when I get that it isn't a thought It is a critique because people will say stuff like oh well Who cares about figuring out all the stuff about programming or? Scripting or configuring a Linux system.  That's a good question now first off There are a couple answers to that the first thing is when I let's say I'm learning something about how to Shell script or something like that that is not wasted time in itself because I am not the kind of person Like that's what I do as a hobby like a lot of people will get home and they'll have nothing to do and they just want to brain Brain drain, you know, so they'll play video games or and play video games for hours Or they'll watch porn or something my equivalent is just opening up my scripts folder and optimizing things and that is a much more I mean, it's not just more edifying. It's honestly more consistently enjoyable than any of that junk So the way I look at it is I never lose time when I'm learning about something like this That that is just what I do as a hobby. I'm glad that I have started this habit Maybe it's a hard habit to form but I have it and I don't have to worry about time wasting stuff outside of that So even if it was useless it still would be no cost, but of course it isn't useless at all So one thing I one implicit thing behind this is a lot of people have the idea, right? Oh, yeah, let's say you spend all this time optimizing a script, right?  Well, the important thing is when you're optimizing when you're doing something really well Even if it's just if you're being a perfectionist about it You it might be an optimization that is trivial in that script But you are learning things that are generalizable to everything else. So I did a video talking about this. What was it? It was the Are you too smart to learn anything? I think that was the title of it where I talked about, you know People it's weird because people will have these two separate questions and they don't realize they're the answer to one another on one side They'll say how do I learn about Linux? How do I figure things out and then on the other side? They will have this sort of concern troll question about why do you care about, you know? Optimizing things and and getting things right and knowing how they work on the command line and stuff like that Those are the same things when you actually allow yourself to Tinker with how your system works. That is how you learn things. That is exactly how it's done It is not a waste of time if you optimize a script and you only get a couple milliseconds out of that If you have learned something that is generalizable and the fact of the matter is especially if you don't know how a computer works You are not able to see all the stuff that you will learn if you just give it a chance, okay? So it you will only be able to see that after you learn it, okay? So I'll just say learning how to script is not like, you know, if you play a video game You are only gonna learn things within that video game with a couple very rare exceptions Okay, you might have some you might gain a little world knowledge in playing a let's say a historical video game or something like that Like that whereas if you're scripting you are gonna learn stuff concepts optimizations You know how a system works you're gonna learn a whole lot of stuff all the time So that is one of those things that is basically never never ever gonna be a bad investment So that's what I say to people like that It's it's just a hobby so it's not wasting my time be you're gonna learn stuff and see I think there was some other C That oh, yeah, the other C is a the the other thing is sort of a personal thing in my life Obviously even if I was not even learning anything I have fame and fortune because I have this you you know I do this kind of stuff so and that's the same thing with you I mean you I'm not saying you're gonna become a youtuber or something like that But there are a lot of ways where you are going to benefit from doing what you're doing that you can't see, okay? So that's what I recommend it pays off if you know how to use a computer It doesn't pay off if you use that time to watch porn and play video games, okay? That's the difference. All right. That was a long answer to that question, but it's it has to be said all right Oh and one note. This is less of a question. Well, I get this a lot What window manager am I using and what distro am I using? I'm using artix Artix is basically just the same thing as arts linux except for it uses run it instead of system D I've done videos on system D before I'm not against it I just happen to use it and it basically has the the same experience as using arts linux So it doesn't really make a difference And what window manager I use I use DWM. I've used nothing but DWM for the past year I know that I mentioned Trying out BSPWM because I thought it could be sort of a in-between between DWM and I3 because people always want me to do I3 stuff. I don't care about I3 I want people to switch to it I think I want to do videos on it to encourage people to switch to to DWM Because I3 I just think I'm never gonna use it again. I don't find it elegant or I don't know expandable in the way that DWM is I vastly prefer DWM Maybe I'll explain what explain why in a video Okay, so other question what YouTube channels do I watch? Very good question I basically don't watch YouTube And especially during the times where I'm making videos. I would probably say that I I Produce more video time on YouTube than I consume which is maybe a little weird The fact of the matter is especially tech YouTube channels. I mean I basically consume none of them I have them all in my like news boat I I watch their updates to see like what kind of videos they're putting out But I very rarely watch it a tech YouTube channel. I gave a recommendation a couple videos ago about a channel Brody Maybe I'll put a link to his channel But he has a it's a smaller YouTube channel that's focused on stuff very similar to mine He's a viewer of mine and he puts off so out some really consistent and good stuff But I mean I don't watch that kind of stuff I'm mostly familiar with you know what he's talking about and things like that But you might find it enjoyable and there are a lot of other channels like that as well But I just don't watch tech YouTube whatsoever I basically don't watch YouTube in general if I do watch YouTube It's for one of two things and this changes every once in a while actually I will say the last YouTube channel I watched consistently was Vargs, which has been off YouTube for like a year That is the last YouTube channel. I basically watched every video or at least pretty consistently most of the videos So the two two types of videos I do watch on YouTube one is like tutorials of like gardening and you know farming and Propagating crops and stuff like that. I watch a lot of tutorials on that because that's what I'm doing right now and the other thing I watch is like historical or theological Documentaries usually like catholic or orthodox ones. That's just something that I watch Like on theology and stuff like that. That's not seeing sounds a little left field if you don't know where I am But but I've only been doing that a couple months. I don't know if I will in the future I but that's what I watch on YouTube But as I said, I don't really watch that much on YouTube to be honest Oh and another thing people ask like, oh, what do you recommend for people who want to start a YouTube channel? I did a video is sort of talking about my experience on this maybe a couple months ago It was like reflections of a minor E-celeb or something like this But I think I'll give you the biggest takeaway Just try just don't invest too much time and make it easy and just make it easy for yourself I actually have a friend which is a good example I have a friend and she wants to start a YouTube channel and she was just Belly aching about this video that she had recorded and she was like, oh, I got to do all this editing and all this You know make it look good and stuff like that and I was just like just upload it People want to see the raw footage. They want to see you I mean she made like cutesy mistakes and you know hiccuped and stuff like that in the video But you know my mindset is if you're if you're trying to get into YouTube be as real as possible I still haven't I haven't made a single cut in this video and I probably won't make one Maybe I will but if I do there won't be fancy effects, you know That is definitely my mindset people really It's a testament to your authenticity if you are willing to upload just whatever No, I don't mean whatever like upload garbage upload good content But you don't have to you don't have to sand off all your corners I definitely recommend just make it easy for yourself and don't expect to get big don't expect to get money Don't expect anything. I got extremely lucky other channels that I've made gotten big all get extremely lucky It's not even if you make there is no labor theory of value in terms of YouTube videos You might put a lot of work into a video that doesn't mean it's gonna get a lot of views. That is one fact of life So anyway, let's see. Let me scroll down. I got a scroll down on my computer here. Oh How do how do I manage my website? Here's how I what manage my website? I Open up an HTML file in Vim and I edit it. That's how I manage my website a lot of people I especially nowadays because you all of these static site generators I don't do any static site generation. That's I don't know a bunch of garbage the what the way I think of it is Well, okay The most I do is I have a template file that I copy to a new place that has you know all the basic syntax of You know the HTML page that I want to put on my website that calls the CSS file and all that kind of stuff But that's the most that that's all I do everything else I do manually I just write it myself in Vim and I might have shortcuts to write HTML in Vim I think that static site generators are a big annoyance especially, you know Because what they'll do is they're really just marked down front-ends for writing HTML So like you write and mark down it converts the HTML It might do this other stuff and it's just a bunch of layers of Confusion that you either have to be perfect familiar perfectly familiar with or you have to know how to debug if you're doing something special I just don't like it. I just like I like simple websites If your website is too complicated for you to be able to edit it in HTML your website is too complicated You need to slim it down and that's what my mindset is Everything I do do that's fancy on my website is just with basic tools I think I mentioned in a video before like on on my main page Where it has like the last five blog entries. That's not fancy scripting. That's not JavaScript or any that garbage It's just a said command or a grep command and then I take the first five outputs of my blog page And then replace that line just with shell commands.  I don't do anything fancy. I don't do any markdown. I just write in HTML. That's it I know there are ways of actually Having a web server that renders markdown. I just think it's gimmicky. I'm old-fashioned. I like the way it's easy It's easy whatever Here's another good question a lot of people ask me about this So this is basically how do I live without a job which I need to put an asterisk on but I'll read this question out I was wondering how you managed to make a living from what you do Feel free to point me to a video if you've already addressed this question I'm interested in alternatives to the nine to five myself But I think I need to learn a niche skill that has some demand I was even considering learning something like computer science from scratch Okay, so here's first off you may know I actually do have a day job But my day job is well first off it's I'm salaried and it's canceled So I'm basically just getting paid because of this corona stuff So I'm basically getting paid to do nothing, but yes, I do have a job right now Although it's totally not necessary for me. I live on the income I have from other places all my salary goes directly from my job job goes directly to my savings account I don't spend any of it So anyway, how do I make money by myself? Now this is one of the I'm gonna have to be a boomer here Okay, because you know I'm 30 now and I look back at how I was when I was 20 and you know I had the same kind of questions worrying about things and I'll just tell you this if you make yourself generally competent in not just one skill not just you know Doing computer science or if you're doing that make sure you're generally capable You generally know what you're talking about if you actually make yourself generally capable You will realize you you can do a lot of things for money. Okay? That's the first off the first thing so I'm the kind of person now I have the mindset of oh just walk in and give them a firm handshake That is that I am 100% that kind of person and I know that's dismissive if you haven't figured life out But it is just not as hard as you think if you If you can if everything you touch turns to gold You will be able to find a job and I think if you're in your younger 20s if you're in college If you're a teenager, I think you need to focus on improving yourself in terms of giving yourself skills I can't tell you what those are I know you're looking for like one or two things for me to absolutely tell you this is gonna make you money But you have to figure it out yourself. You have to just Focus on yourself and focus on improving yourself and it will come like it will come and I am of course I have YouTube I didn't expect to have YouTube five years ago.  I ran with it I got lucky with this one thing in life I could have gotten lucky with a lot of other things in life instead and I did that and it's been a pretty stressful or stress-free Thing that's gotten me enough income.  There's my shed at my house is on the other side I took this I can make lots of money just putting out videos and doing stuff online That is not something that you could do, you know, easily 40 years ago You have it you can live in a place and not spend that much money and still be connected to the internet and make a lot of money, okay So it's easy to live without a nine-to-five job if you're generally competent person. That's what I say and yes when you are young My recommendations would just be you won't be I take a job that doesn't Take up all your time so you can tinker with the things you need to And don't go into debt. Absolutely.  It is nice in my situation because I only I deliberately chose a public like in the state that I went to I went to College in Georgia in Georgia they have a Scholarship or basically if you go to a junkie public school and you have grades that are fine enough They'll basically just pay for your education and I deliberately chose that rather than going to some swanky expensive private school Because debt is a permanent liability, right or if you have to go into debt to go into go to school Don't go to school. That's my recommendation. I am so I'm very happy whenever someone tells me I was thinking about going to college Then you told me it was a bad idea and I think I'm having second thoughts because college is definitely The only reason you should go there is to get some kind of rubber stamp Don't expect anything from college other than that a very expensive rubber stamp So that that's what I would say and I was gonna say something else. Let me think about it Shoot what is it? Hold on. I gotta think for 30 seconds Okay, I remembered I had to make a cut in this video, but I remembered Minimize your expenses in addition to just getting generally competent you want to live on as little as possible I don't do budgeting because I just never buy anything that that is one of the biggest things because I'll tell you I went to graduate school and in graduate school. You have so many of these bratty incompetent Rich kids who do things like they'll they can't cook for themselves or they if they do that's like a meme Oh, I'm gonna cook tonight because usually they go out and eat at restaurants two times a day They spend like 30 bucks on that they spend they have terrible habits like coffee or cigarettes or something like that That's like 10 20 dollars on that sometimes and it's insane the amount of money that people spend through and you need a I mean Even when I was a teenager before but when I was in that state where I was like how many of how am I gonna figure out What to do well at least I can minimize my expenses at least I can say How little can I get away with spending every single month and it's not an issue of budgeting I know a lot of people like budgeting like budgeting and they find it useful But I am just like I am only gonna buy something it is if it is the last Circumstance it if it's the last thing I have to do, you know what I mean So that is another recommendation. You just got to do it you It's insane how much money Like in the thing is if you are not go it's and it's crazy to go out to restaurants all the time Especially if you're alone, that's pathetic going to a restaurant alone if you go to a restaurant It should be an event with friends and stuff like that But just don't spend money you don't need to and every time you spend money You're relying on someone else to do something you should be able to do you should be Practicing your skills and cooking or doing something else or doing basic things for yourself And that's one of the great things about living in the middle of nowhere People have this weird idea that you're gonna be spending more money somehow and in fact sometimes things are actually more expensive But it makes you more independent. You start buying in bulk. You start planning ahead You start having a life where you spend so much less money Even if you actually move to a place that weirdly enough you might be paying more for which will nearly never happen All right, so that that was the last of the questions. I feel like that last one is worth its own video Maybe I should just do that but anyway, that's it and I'll see you guys next time Congrats to everyone on a million hours viewed consumed. See you guys next time


I cannot see. Did you make me, do you want me to be a co-host just for safety reasons or?  I'm a co-host. Yes. Absolutely. Okay, so I think if everybody's ready, let's start. So welcome everybody to this basic notion seminar that we are having. And I'm very happy to introduce today Professor Giovanni Bellettini from the University of Siena and of course also ICTP, who will speak today about a very nice piece of differential geometry, namely the mean curvature flow of closed hypersurfaces. So the floor is yours. Thank you. Thank you a lot for the introduction. So this is a basic notion seminar. So I will try to be as elementary as possible and keeping the notation and the technicalities to a minimum. So let me first define what is mean curvature flow. And then I will discuss some examples. So the definition can be given in various ways. I will use the definition without the parametrizations, which may be simpler. So let ET be a family of solid sets contained in a REN. So for any T, which is a parameter, which is time ET is a solid set. And solid means say the closure of an open set. And I say that we say that T in ET or say T in the boundary of T is the same. I will somehow speak about evolution of hypersurfaces. These hypersurfaces are boundary of something. So for T say in some time interval, zero capital T, we say that this is a smooth mean curvature flow, which means that at the end the surface is the hypersurface will move by mean curvature if we define the following function. So this is the distance function with the sign is negative inside the solid set and positive outside. So this is my set, this is ET at fixed time. If this function is smooth, is smooth in a neighborhood in an open set containing all the boundaries. So which means essentially that the flow as a function of time is smooth and then also each boundary is smooth. So this is the following equation holds for any time. This is the PDE. This is the time derivative. This is function. This function is defined everywhere actually is smooth in an open set containing all boundaries. This is the gradient in a REN. This is the Laplacian in a REN. So this is say if you want the heat equation, this term here equal to this, but for the distance function with the sign and the gradient of D is the unit outward normal on the boundary. This is called mean curvature. Sometimes denoted also by H is a scalar mean curvature and is equal to the sum of the principal curvatures. Notice that this is a PDE, which is however valid on an hypersurface and this hypersurface is moving. Okay. So this is just valid on the boundary. So this is, this is called the normal velocity. So at the end, I'm saying that the normal velocity at each point is equal to the mean curvature. The mean curvature vector. So which means that if I have an initial set, say initial curve, so this is the interior, this is the boundary. It means that it means that at each point, the velocity here, the convention is that the velocity is where the curve shrinks if I am locally convex and here instead expands locally. So the evolution here, the normal velocity here must be proportional to the mean curvature here. Okay. And this must be valid at all, at all points of the curve. Okay, why I have given this definition. Well, this definition is you see there are no parametrizations. This is convenient in some sense. But there are other ways to define. So this is a remark. So there is a remark here, which says that there are other ways in particular. The flow can be defined and studied describing the manifolds instead that set of points using the maps immersion or embeddings of a given manifold using parametrizations. So I will not use probably here. This language, which is very convenient. For at least two reasons. Because it can be useful to treat immersions, not only embedding. So this means that my flowing manifold could self intersect. I will avoid here this case, but in general using immersions. This can be done. And another reason for which it is very convenient because it takes advantage of the covariant differential of the covariant language, covariant derivatives. Okay, so there are at least two ways to describe that this is maybe, maybe simpler, but maybe not not always the most convenient way actually. Okay. Why this, this problem. So another remark. This is this evolution is the fastest way to decrease area of the moving manifolds. So this means that in a sense, this flow, try to evolve this curve, say, as time is moving is going on. This tries to reduce the area of the flowing surface as fast as possible. This is not so easy to not so easy to make rigors, because typically, I mean, people says that this, this is the gradient flow of the area functional, which essentially would mean that this is sort of generalization of in. If I had the system of ordinary differential equations. This is the gradient flow of F, which goes trying to go towards a minimum of F in the fastest way. In this case, this is the analog of this but the difficulty are several because here we don't have say a vector structure. Maybe, and it's almost it's very difficult also to say what does it mean gradient of the area functional in difficult to make rigors also what is this for F area of some surface. This can be made rigorous and if you want to take to to have really a rigorous way to to understand this, this can be for instance, obtained from an algorithm by Algren Taylor and one in 1993, which is a sort of, which makes somehow, which gives a sense to this idea for when F is the area and uses a sort of implicit Euler scheme for the mean curvature flow. Okay. Examples it is always bad, very good to have several examples here in mind. So let me give you the basic examples. First example is is suppose that this is a minimal surface, which means a smooth hyper surface with zero mean curvature. So this is a static solution of mean curvature flow starting from Sigma is from this itself, because this is a zero mean curvature so zero velocity doesn't move it remains fixed and this is a solution which doesn't move. This is a very trivial example but very interesting because it could give barriers. Example to here suppose that you have an initial set, which is the interior of a solid ball in a rain. Okay. The solution a posteriori I mean a solution but at the end the solution is still given by shrinking sphere where r of t. So this is called where r of t is this radius here so this this fear as constant mean curvature. So it shrinks remains a sphere remaining the same shape and shrinks to a point here. You see, this is interesting because shrinking to a point can be considered an example of singularities. Of course a singularity should be defined but this is a moment in which the curvature blows up. And this moment is when t is equal to r square divided two minus one. So let me call this T max. So this, this evolution is defined up to T max actually. So T is from zero to T max actually from minus infinity to T max but for the moment, think that we started from our zero. And then we shrink up to T max and this T max. So the third expression of the T max here you get that and you compute the length of the second fundamental form, which is the sum of the, in this case they are all the same. If you compute this, this turns out to be a constant. T max minus T. So the length of the second fundamental form that is the square root of the sum of the squares of the old curvatures and this constant is one of a square root of two. In any case, you see that this blows up when T goes to the T max with a certain rate. Example two, example three is cylinder. So you say S one times R. So suppose that you start from say B or zero times R or more generally by you start from S M times R and minus one minus M. So you have a cylinder and the cylinder shrinks. This is non-compact by the way. So there is a serious difference between surfaces which are compact and many forms which are not compact. This is a non-compact case. One has to be very careful treating non-compact initial conditions. But at the end, if you have a cylinder, this shrinks remaining a cylinder and you have that this evolves with RT and this RT is the same as here and you just substitute this with M. And you get the same. So at some time there is a singularity and what you see is sort of the cylinder collapse to align to its axis. At some time and you have for similarly a blow up of the length of the second from the mental form. There are other important examples here. Other important example, there are very many other solutions. So example. For So these are graphs. They on R and minus one. So you can suppose that your initial condition is unbounded. So you have a graph over an entire graph over R and minus one. So you are still in the non-compact case. It's delicate, but you can at least have a you can at least say what model this using just one parabolic PDE non-linear PDE. And then you have with your graph initial. So the boundary of the initial condition is the graph 0 say given and this can be studied and it can be proven that under some conditions. There exists a solution and it remains a graph. So the property of being a graph is not destroyed by by the the the mean curvature flow. Okay. This is another geometric factor. Next, the next examples. Next examples are given by the so called the homothetically shrinking solutions. For example, five homothetically shrinking solutions. I don't want to enter here in the details. So this is a very big subject. Let me tell you that the sphere is a motetic why is a motetic because its shape remains same up to a rescaling and the same for the cylinder. So you can try to look for special solutions of. So you mean you have this PDE and you look for say special solution with time and space maybe independent. So special solutions. Solution where specials mean which are say is an homothetical a given initial condition. Several, there are several solutions. Actually, this is an important subject because understanding. Let me tell you one thing. So understanding how these solutions is strictly related on how this singularize. I have to define what does it mean but I as T that goes to T max in some sense. So this is for the moment, not rigorous because I have not defined what is the max yet. I don't have not tell you that there are singularities but that we just point out that the study of this special homothetically shrinking solutions is interesting by itself and also because it gives you information on how possible singularities may form for time slightly smaller than capital than T max. For instance, here, this shrinks to a point like a sphere this shrinks to a line, etc. In particular, there is an interesting solution here. Interesting solution. Similarity which is called the Anganand. Taurus, which is sort of so you have your. This is your the solid torus is your smooth initial condition. And there is a special torus, which remains, which is has the same shape and this shrinks. Remaining torus. Okay. Notice that for fat torus somehow or a torus of the sort to one can imagine that here.  Maybe at some time there should be something strange here on this point. And for a thin torus one could expect that instead here you see in torus means that here the mean curvature is very, very high toward the inside. And so this means that one can expect a normal velocity very quick toward the inside. So one could expect that before, if the torus is long but thin, one could expect that this total could collapse somehow to a to a curve here before disappearing. Okay, so all this kind of studies related to this to this could be related also to this to this problem here. So let me tell you now some result here. So the first theorem obviously, which is not trivial at all is the local existing. Say, if is smooth and compact, then there exists a unique mean coverage of flow, starting from, from this initial condition is smooth mean coverage of flow, starting from this initial condition. There is a positive time for some positive time. And actually, we can say also that we can we can arrange things so that there exists a maximal on a maximal time interval. So here means that you cannot go on after the max that you are you're not smooth anymore, I will be more precise in a minute. What happens to the second fundamental form as when I go to team max but this at least allows to start to the flow. So we are sure that there is a unique solution or smooth up to some maximal time. So here's the theorem is not trivial the first proof maybe is. I don't know, but I think it's gauge Hamilton 86. Then there is a proof by Evans. Prack, using a completely different proof of a completely different language 92 then there is another proof by whiskey. I think. Okay, so this is the first local in time existence result. Let me tell you a remarkable property of this flow. Which is which I will try to describe remarkable property of comparison property of comparison. Suppose that you have an initial solid set smooth with bound say compact with the compact boundary. Suppose that you want to do to smooth initial solid set so one inside the other say one is this, which is inside the big sphere I don't know so. So this is the one at the initial time this is two solid sets one inside the other, and then take their unique mean curvature flow. So for all these times these capital T is positive this is what it starts from me one this is more than starts from me to they are uniquely defined by this previous theorem one. Okay, then if initially one is inside the other. The same is preserved. For all. Times. This is actually very strong and important comparison principle is an important tool. And it's called comparison principle. So let me try to tell you why this this should be expected to be true. I will not do a proof because there is no time enough but I can at least tell you something which could suggest why this is true. Which could suggest. Why this is true. So let me draw a P and it's a simple picture suppose that we did we are in the case of the plane so we have the one and we have say it to and the one initially time zero is contained in it to suppose for simplicity. And the boundary of the one is at the positive distance from the boundary of it to just for simplicity. So we are not actually in this picture but we have say something like this. Okay, so you want. For simplicity. Now. We have a number of compact curves close the curve send. So that there are at least a pair. Maybe many pairs, but in this picture just probably one pair, which minimize the distance between the two boundaries along the flow. Take suppose this is one T to T for they they are once in its initially there they are one strictly inside the other the flow is smooth so for small time there, of course, still one inside the other. But what happens for larger times. So take a point a couple of points here, which realize the minimal distance between the boundary of. So let me call the of T distance. So this D of T is the distance so it may be realized by several pairs here in this picture. We have just two pairs probably. And the idea is to show that actually a statement stronger than this, not only they are remain inside about the distance is non decreasing. The idea is to show that the T. Is non decreasing, which means that for larger times these shrinks and this for this this quantity is maybe larger. Okay. Now, we have to be careful because this is a very simple picture but you have to imagine something more complicated with those torus inside the other. So I mean, this is just a very simplified picture. So the idea to show that this this is not increasing is to find a different I mean to show that the prime where it exists. Supposing it exists. To show that the prime is larger equal than zero and two and one actually one shows that if you complete the prime. I mean it's not actually the prime because this the prime could not exist but this is sort of right derivative, but essentially the prime. One can show that, essentially, this is given by the difference. I mean this is these are two points which realize the minimal distance is not difficult to show that the tangent plane here and tangent plane here must be parallel. So this means that the normal is the same parallel they are parallel the normal. It is possible to show that this is equal to be one minus V two very one say V. I for I equal one two. So this is the difference of the two velocities. So but this is actually h one minus h two because they have the evolution equation so the velocity is equal h one and this is equal to h two. So why this should be positive or negative say. Well, the idea is that you think that now this is a point of this minimize the distance or if I make a translation here, rigid translation of this set and they put in contact. Let me translate this rigidly and put it in contact. So they have the same normal. So actually what I see locally here around this point is the following situation. There is a manifold here and another one which touches and locally. This is the largest and this is the smaller. Now, if I, I mean, the mean curvature is invariant under rotation and translation so I can do I can put arranged things that this such that this is horizontal. So this is somehow horizontal this point. And then and then if I compute the mean curvature at the point where I'm locally the graph horizontal graph like this I compute the curvature here. I mean curvature turns out to be the Laplacian of the function which described locally this is a graph. So at the end I have Laplacian of you one minus Laplacian of you to this only at the points where I am where the tangent plane is horizontal. And then this is a local I mean if I take the difference. This is the Laplacian of the difference and this is a local minimum and it's immediate to check that using minimality for a function that should be non non negative definite and you are done. So this shows that so this should be expected this is not the proof but maybe the idea of the proof. And at the end is that we can compare them in cover show of the set with the big with the mean cover cover show what another set which is inside at the contact point. Okay, so this is an important thing and then let me tell you. First, remarkable, I would say a difficult theorem. This is split into two because historically it splits into two. So, let me tell you your M three. This is gauge Hamilton. X. So suppose that you have you are in two dimensions so this is mean this is curvature flow shortening flow, a curvature flow of curves. If you have a convex say smooth suppose that the this is smooth and convex and bounded. And then, then its evolution flow exists and remains is convex remain convex for all time. So the max and at the max. At the max. The curve shrinks curve shrinks to a point. So nothing can happen before then shrinking to a point like in the case of the sphere, the curve shrinks to a point and once rescaled. It is close to a circle. The start from a convex curve what I see of course the evolution is not explicit so but it remains convex there is a magic point where it must shrinks and nothing strange happens before and here. What I really see is sort of almost a circle which shrinks and the circle is remember is a self similar solution. Okay. The theorem. Has been improved by remarkable. Theorem by Grayson now I don't remember exactly the year. Come here later. Grayson, who shows that actually. Grayson was able to remove this assumption. Suppose that now your curve is compact as mood. But not convex. So, this means that my initial curve. Can be very embedded, of course, but very embedded sorry maybe this picture makes not so clear it is embedded. So this zero. So what happens here is able to show that it remains smooth. Remains smooth. Of course the max is finite and all these cases are. So it remains smooth. And at some time at some tea. This is becoming convex. And so, once it becomes convex. So it's able to prove a say global existence theorem in the sense that. If I start for any smooth compact embedded curve. This doesn't do any, any nice, any nasty thing. I mean, it remains smooth at some moment becomes convex, then it remains convex, then it shrinks to a point that becoming a once rescale the circle. This is really remarkable because you see, you can imagine that the curve. Can be initially pass very close to itself, even if not self intersecting but. Can pass very close to itself. But still and one could imagine that at some moment that could be self intersections and this cannot happen by the theorem of Grayson so it is embedded remains and bad that nothing stage happens and this is the first. Now for this theorem there are more than one proofs. There is what the original proof of Grayson then there is a there is a proof by whiskey, which for proving this passes through some classification of singularities. So also this proof is difficult. I mean in some sense. Also this is not easy at all and then there is a more simple proof I think by Ben Andrews. Another proof. Okay, now let me tell you another remarkable theorem here. Another remarkable theorem. So this is what concerns so you see in two dimensions for cursor actually singularities is the singularity of the of the sphere of the circle. In some sense. To see more complicated singularities in the plane one should start with the, as immersed and not embedded curves. So one should start with something like this say, or something like this. Of course if I want to flow this I have to abandon the distance function I should go into the language of parametrizations looking at the maps this is not a singular point anymore for the map. And it is, it can be proven this has been done by Angon and that this is so small if you if you arrange thing that this is sufficiently small then it shrinks very quickly and forms a cask. The cask and another kind of singularity but I cannot discuss this here. Second theorem. This is a remarkable theorem. The four. This is whiskey. 1984. Now suppose that I have in larger dimension and suppose that by initial boundary. My solid set is smooth and strict and strict and strict to convex. And then again. Exist is smooth. Quickly convex and again it leaves up to. Some T Max and after rescaling again properly scaling, which means that one keep keeps the area of this boundary constant essentially here after rescaling. And then it approaches to the it's very close to the to a to the ball. This one. So again what we see here we start from a convex set convex body say the theorem of which can say is that this shrinks it remains convex it does not do anything strange. And that's a moment. This is singularity. Time. Now the proof of this is different. Of course a strictly convex body surface as also is also mean convex clearly. Not only I mean strictly complex much more than, than this. And to show that this condition is preserved is not difficult. If I start. This condition to show that this is this is preserved is not difficult because it follows by some evolution equation. Some evolution equation for the mean curvature itself. And using this parabolic PD. This is sort of Laplace. This is called the plus Beltrami operator on the moving manifold. But in any case is sort of it equation for the mean curvature plus a nasty sort of term here. Um, but it's much more difficult to show that strictly convex it is preserved because this is the positive definiteness of the second fundamental format only its trace should be remain positive but all the second fundamental form should remain a positive definite. So to show that. The second fundamental form. Remains. Positive definite. This is more delicate one has to show one has first of all to find the. The evolution equation for the old elements of the second fundamental form which is a sort again is a sort of is a sort of something like this then there is minus. Sort of equation like this. So not only one has to find this which is ending down by whiskey. But also one possible way to prove is then once you have the PDE solved by the second fundamental form and all covariant derivative of the second fundamental form you have to show that you have to implement the maximum principle for 10 source by Hamilton. This is one possible proof. So it's, it's, it requires a lot of work. It's, it's whiskey 984. Yes. So these are various remarkable theorems. And now let me show you that the singularities in general appears singularity can appear. So, oh, let me tell you an example convincing example. Which has been proved in various ways. Now I tell you the idea. But this is an example due to. Gray zone. So, you see, we start from a convex object. This does not single a rise in the sense that it's singularized at the max, but at the max also the surface disappears. So it's a sort of not too complicated singularity. And the same for curves, even if they are not embedded, but not, but not not convex initially. But now Gray's on argued that suppose that you have they in three dimensions. So this is a sort of dumbbell. We're here. This neck is very, very thin. So here the the mean cover sure is very, very positive. Here is positive, but maybe not not large. But they have two balls essentially almost joined by a long and thin neck. And the idea of Gray's on was the following was that I want to show that this what could it mean this? Well, I could show that we have there is some time at some positive time. So let me call T max at the max. One possible way to say is to say this is that say the linear for as T grows to T max of the volume of the solid set. So this evolves in some way. But if I arrange things to be this very long and to be this very large this very long and this very large, maybe he's able to prove that at the max the surface is not disappearing. Something remains in the sense that the volume inside is positive. And the idea is the following. The idea is that here I can put a big sphere somehow inside and this sphere evolves with its evolution law. And we know by comparison that as soon as the flow is smooth the sphere and of the object outside the dumbbell this must remain inside. So it cannot happen. I mean this this dumbbell must leave for it for some somehow for a time sufficiently enough that this is still leaving. On the other hand I can put, well here there is some argument different from Grayson. Here I could put say an Anganon torus around here. Okay. So I surround I surround the neck with the torus the Anganon torus which I know that it shrinks to something but maybe to a point but it does not synchronize before and then he has some time. So if I arrange things so that this this you see this neck must remain inside the hole of the torus as soon as everything is smooth. And so now you can morally intuitively believe this takes some time to shrink and if these balls are sufficiently large it takes more time to shrink and so this cannot disappear. And so what happens here at the end there should be some some some similarity before. So at the max before that the the wall dumbbell has disappeared and this can be made rigorous in various ways and it's been done by Grayson himself. So singularities appear. Okay. There is an interesting theorem here theorem by whiskey theorem. I don't maybe five or six five. And suppose that this is mood. Compact then we can detect some out in max in the sense that so T max is maximal time of of existence then the limit be converges to T max of the infinity norm of the second fundamental form of the manifold close up. So we can detect T max in the sense that the max necessarily something strange happens in the sense that if I compute the sum of the square of the principal curvatures then as soon as T converges to the max from the left this sum blows up. So this is quite I mean it's not easy at all. I mean it's not an easy proof it inspects the the various PDE sold by the in the sense in a sense why this should be true. Well if the second fundamental form does not blow up if say remains remains bounded somehow it is possible to show that not only it remains bounded but also all covariant derivatives also remain bounded in the same interval and then once you know that all derivatives of the second fundamental form remain bounded it is possible to take the limit of the manifold as T goes to the max you can go take the limit of the manifold you have a limit manifold this requires some work but there is a limit manifold for which you can restart the flow after start the flow on a small time interval T max say T max plus delta some positive delta and then this contradicts the definition of T max which was the maximal time of existence so and indeed I mean this is a starting point I mean then the story goes on a lot still big work by Wiskon and others but so the rate say the rate of the rate of so how quickly the norm of the second fundamental form one looks at the rate of blow up of the length of the second fundamental form as T goes to T max and then this through the allows to start to start the study of singularity formation like in the the Grayson example of the dumbbell study of singularity formation allows together with the so called Wiskon with the Wiskon monotonicity formula I cannot tell you what it is now I have two ingredients to start the study of singularity formation and try to classify how singularities look like and their relation with the homothetical shrinking special solutions that we have discussed in example three or four okay so now this is a very interesting subject let me tell you now I have three minutes to conclude by saying that singularities are maybe the most interesting phenomenon here so but in singularity here one has to invent the notions of weak solutions which means that solutions which are global defining time so they go beyond the first singularity time second singularity time etc so they are globally defined look for weak solutions so called weak solutions globally defined also in the presence of a singularity so where the Nincor Vashur is not defined point-wise say for instance globally defined for all times okay and of course coinciding with the smooth solution as long solution as long as it exists so there is a sort of extension of notion of flow which for times before the Tmax is of course the same as the classical flow but after Tmax it is still defined and it goes on the process goes on and of course I mean this is another subject which is quite huge try to so there are I think at least maybe 8, 9 or maybe also 10 different notions of weak solutions weak solutions what was given by Braque historically is first one is given by Braque 78 so the subject it's an interesting thing probably is to study how this the property of this weak solution and how these solutions are related one each other because they do not coincide in general after the formation of singularities so how do they relate each other unique I mean and so sort of things like this okay I think that my time is over and thank you okay so thank you very much for an excellent talk it was very interesting and also very nice to see an actual Blackboard talk first time for me in a few months so let's see if there are any questions you can write either in the chat or you can raise your hand in the participant list and they will call up on you one by one so let's see are there any questions okay so while I have a question? yeah okay so we have a question so Shal Faisal yes you're welcome to ask your question please so in the initial remark about the torus the defending on the initial picture the genus of the torus can disappear so what happens to the genus of the general genealogy surface what happens in general to a genus of the now this I don't know what happens in general to a genus I don't know I think well of course you again you need the notion of weak solution because if you imagine that the holes now disappear in a sense that if you have a fat torus say then this singularized at some moment here and then maybe after now here you need the notion of weak solutions because you want to go on in order to answer to your question so what happens depending on which solution you choose maybe okay this becomes a sphere and then it shrinks so I think that here you need the notion of solution this so in general I cannot tell you there is a there is a study on this but I'm not completely sure so I studied by Brian White which I don't know I don't remember now the results but as far as I know maybe it's Brian White which studied the partially maybe this problem but of course you have to select the notion of weak solution so for this maybe he was using sort of set theoretic solution or also viscosity solutions maybe but I cannot tell you more than that right second question you define this distance function I think in theorem 3 can you please draw some picture to show that it is not in general smooth yes the distance function has a sign in general the distance function from a manifold of course is not smooth exactly on the boundary of the manifold and this is a problem no so if you take just the distance just the distance of the point so f of tx just the distance then this is of course say you have this curve here so this positive here this zero here but the point is that I want a PD exactly on this manifold here where this distance is not differentiable you see because locally here what you see locally here your positive here your positive and locally you have a corner the graph of these exactly a corner on your curve so exactly there where I want to have a PD exactly on the moving manifold the distance is not smooth but now so this is no I cannot use it but I suppose that this part here make it negative okay make it negative this means that instead of having distance positive everywhere I have negative positive and zero and now it is now it is possible to show that here in a neighborhood of the curve the this sign distance function which is now positive and negative and zero on the manifold is smooth as smooth enough in a sense that smoothness of local smoothness of a manifold here is equivalent to say that this distance function now with the sign you see when I am inside this is this is zero so the distance is negative when I am outside this is zero so the distance is positive it is zero on the boundary so it is possible to show that a manifold is smooth if and only if this function here now fixed take a fixed for a moment there is a neighborhood of your of the of your boundary in which this is smooth so this is equivalent and so this is essentially a way to describe your manifold without parametrization just as a level set the single level set of a single function globally and this function is very very nice because it is a distance function so where it is smooth so in the neighborhood of the manifold you have the a-icon equation and so this a-icon equation is essentially a way to extend the normal vector field keeping length one not only on the move on the manifold but also in a small neighborhood and once you have this extension of the normal vector field in an open neighborhood all curvature invariance all curvatures can be described instead of using parametrization using the derivative of the distance function for instance this is equal to this is a theorem so on the boundary the mean curvature is the Laplacian and the advantage of this that this Laplacian is a standard Laplacian in a so what do you do you do the Laplacian in an open neighborhood and then you restrict on the manifold and the restriction is equal to this stage and so on second fundamental form can be described by the action of the distance etc etc right thank you okay thank you so there are two other people who have questions so the first question is from Sog Gatumov so if you can unmute please and ask your question let's see if he is here okay otherwise maybe we wait for that and I will let Hamza our own diploma student Hamza honestly ask a question please okay thank you for the talk can you hear me yes sure okay so my question is whether we are these results obtained are considered only for manifolds embedded in the acridian space or the results are still valid when changing the metric and the second question is in this case whether changing the metric I mean whether there is a surface of R3 that may shrink to a singularity for a specific metric but not for another one I mean whether changing the metric may let us avoid singularities in some way okay what I know is that this so first of all differential geometers usually do not use often do not use the distance function they prefer to use parameterizations so they look at the whole map instead of just the image of points but this is an advantage that they can treat also the self-intersected case the mean differential flow has been studied by whisker and others also sitting in a Riemannian manifold so and there are there are a whole literature on short time existence singularity time, team max, etc. also in a Riemannian manifold and of course maybe there it could happen I mean not necessarily true that maybe you should disappear in finite time it could probably happen that you go toward a sort of geodesic I am imagining that suppose now that you suppose that you flow an initial curve which is here on the boundary of this surface by mean curvature and then you flow and then maybe it happens that not necessarily you can shrink you could go maybe toward a sort of geodesic curve there so yes so the answer at the end is yes there is a big literature I think on the mean curvature flow in a Riemannian manifold in general I don't know if I can answer it to you but thank you okay so did you have any more questions Hamza or that was it the question was I mean in this case now whether a surface for example the sphere can shrink for example the intrinsic matrix we have seen that it shrinks to a point following the associated mean mean curvature flow now if we consider the general case whether it depends on the choice of the matrix this fact or not I mean whether it's always the solution is that it shrinks to a point may depend as I told you that this does not give you an answer no? I don't see how I mean in this case we will not have the same result as before I mean in general this probably could evolve toward this and remaining remaining here now trapped here ah okay let's see okay thank you okay so finally going back to the question from so again to most of microphone is not working but I will ask for the question instead so the question is does the weak solution suitably defined exists globally in time yes as I said there are there are several notions of weak solutions with which ranges from definition from geometric measure theory to purely PDE's definitions one very popular is the viscosity solution which is a notion of weak solution just maybe uniformly continuous function which is for which essentially this PDE one studies this one so this is the general non-linear PDE so one can give a notion of weak solution not necessarily see to actually only lip sheets to this PDE geometrically each level set of this possible disputative solution evolved itself by Ming curvatures not not only one level set but all together they evolved by Ming curvatures so there is a big theorem here existence and more over more difficult and uniqueness uniqueness of a viscosity solution for this Ming curvatures flow problem and this is goes back to Evans and changing agotto in 93 and also Evans the year I don't remember sorry maybe it's 91 Evans at the same time famous result for this notion there is a global existence for all times uniqueness which is much more difficult in this case and this gives a solution which coincides in a proper way to the classical solution as long as the classical solution exists so it is just one example there are actually historically black in 78 was the first to define a notion of weak solution completely different from this using very full theory so geometric measure theory is a sort of okay very full theory gives another notion of solution for which uniqueness has maybe is less but in any case there is another theory of weak solution and then the story goes on as I said starting from black passing through changing agotto Evans then Ilman the Georgie and others give several different notions of weak solutions okay thank you same participant also the second question which is the notion of mean curvature flow safe for space like hyper surfaces in a Lorenzian manifold so the Lorenzian manifold what I know is a sort of analog I mean in a Lorenzian manifold I mean with the Minkowski metric say what I know is that the analog of this problem is in this case now time has a different meaning they are not any more parabolic problems but are more close to hyperbolic problems and what I know is that sort of geometric similar problem is a minimal surface equal to zero where now the mean curvature is in the Minkowski space and so I mean time and space they are somehow the same role and so they speak more than like minimal surface in the Minkowski space rather than mean curvature flow so the analog as far as I see of mean curvature flow is minimal surface but in Minkowski case this is what I want to say and it is not far to some which is not any more velocity equal mean curvature but this is sort of acceleration equal mean curvature times a factor which I think if I am not wrong makes the equation Lorenz invariant but this is a way to see, to look at this is mean curvature of the time of the time slice all is like to say h equals zero in time space there is a big literature on this I think they are called classical strings ok thank you very much for that answer so I think these were all the questions we had for now so on that note I would like to thank again the speaker Professor Bellatini for a wonderful talk and also the camera crew the technical staff for making it possible so I am for all the participants for being here and so before we leave I just want to start a small poll but before you leave I would be grateful if you could answer it takes just 20 seconds and otherwise I hope to see you see you next time so thank you all thank you Professor Bellatini and if you are a diploma student you can also stay if you have some more questions I think this is ok