So I'd like to welcome on the stage a bunch of people from Polymer engineering team to answer all your questions So first Justin Fagnani is an engineer on the Polymer tools team Monica is an engineer who you've met many times today and yesterday on the Polymer team Rob Dodson developer advocate extraordinaire expert in all things web and general internet celebrity Wendy Ginsburg the product manager on the Polymer team itself who gave the keynote yesterday and Steve Orville the mastermind or one of the masterminds behind the Polymer core library and engineer on the Polymer core team as well All right So we have two microphones One microphone up here two microphones up here So please come up ask your questions. This is an open space. This is a safe space.  Please come up ask us anything You can also tweet at us at the hashtag ask polymer and we're looking through those tweets live And so I'll be I'll be looking through a machine up here to try to get some of those questions delivered to our panel as well All right Great. So to kick us off. I have a question in the barrel Which is that? Lots of exciting stuff coming out yesterday and today with Polymer and MPM and ea6 modules Lots of excitement about how to develop components also a little hesitancy for a lot of folks who really enjoyed being able to author HTML in HTML So is there anything that you're thinking about in terms of how we can kind of better support that use case that has been so special to So many Polymer developers of authoring HTML in HTML Please Yeah, so I mean I first want to say that we hear you We spent a lot of time yesterday and today talking to people who were you know kind of lamenting the loss of Authoring in HTML and it kind of makes sense because that's something that's been a little bit unique about Polymer and You people have chosen to use Polymer and that's why you're here because you like it a lot And so it's definitely got us thinking about ways we can you know help support that kind of style and that kind of workflow Even as we move to JavaScript modules It's important to note though that also our philosophy of using the platform kind of means that like when the platform gives us a Tool and in this case kind of denies us another than one that we were hoping to use The we our choices in a sense made for us a little bit We have to go to JavaScript modules because that's the native loading solution on the web but we can look at ways to kind of layer the project and Provide ways to you know Have some tooling or something like that The important thing is that Polymer 3.0 is very very early in preview stage And so there's a lot of time for feedback and a lot of time for experimentation with solutions to give you the You know the ability to write HTML in HTML The same Yeah, I mean the actual string that you write is the same template. In fact, you could draw a line from every line in a HTML import to a line in a JavaScript file and there would be a one-to-one correspondence between every line in there We're just following Sember and that's why you have to bump the major version. Yeah So we hear you we have some interesting ideas on how we can You know serve the best of both worlds Yeah, I just add quickly that you know again based on feedback The reality is Modules are here today and we want to support them You know, but there's some spec work potentially to develop an HTML extension to those modules And that's something that we'll be thinking about and you know Evaluating and seeing kind of you know if our users are really clamoring for HTML On our own team, I will say that there was definitely a bit of like difficulty ripping that band-aid off But we're also excited about some of the new things that modules are gonna provide. I mean lit is an excellent interesting direction and You can't do that in HTML. So You know, we're gonna be experimenting and listening for feedback So a closely related question that should hopefully be quick if I'm working on a new project right now Should I target 2.0 or 3.0? 2.0 definitely 3.0 is very very early and the other thing to note is that the When we published core and all the elements those were literally converted automatically by a tool minutes before we published them up It was like one process So if you're on 2.0 and you keep pushing forward and you do that we're gonna take you to 3.0 automatically In fact, you can tell that they were published while we were here because we Got rate limited by github on the conference IP Wi-Fi so that if anybody was trying to use the github API you couldn't do it because Justin did all of it No, we used to get up token that wasn't us. Oh, there's just so many people in the code lab Yeah, I think you know a little bit of straight talk to steal Gray's term is you know We're really aiming as Justin said to earlier to you know, not change the API between 2.0 and 3.0 Just as little as we have to absolutely have to you know, and actually Kevin's app that he built He transferred the whole thing using the you know the tool that Justin's team has built And you know it had some bugs for a while But you know the latest version works pretty seamlessly and the kinds of I mean, you know I think Justin sort of obtusely mentioned we're changing as little as we can or maybe was Fred We're changing as little we can honestly like give me give you just one example. There's the Polymer import href What did that do it created an HTML import? Well, there's no more HTML import So you have to do that differently, right? This is like about what we're gonna change. That's the only change Yeah, and in fact this this brings up a good point You know if you're vending open source elements out there We actually recommend that when you use the converter you do not change the API of your elements at all Treat them is having the same major version even if you bump the major version because it's now modules Because that one that means when your users run the converter, they will automatically be compatible with your element So we look at this transition is is not really an API breaking change. It's just a repackaging All right, let's go to a live question I have a question regarding tooling and improving developer experience. Is there any opportunity to you? And you were developing and to have provided hot module replacement Like you are coding and you immediately see the changes in browser hot module reloading. Yeah, that's a tools question. Yeah, I mean We we don't have any work planned on that There are several ideas on how you can do that One of the things with hot module reloading is you often get into a problem where if you're replacing some JavaScript It's hard to huh It's hard to replace the state that might have gotten in a certain, you know states through some path of code that you took So I was just talking with some people today about ways we could make templates auto reload In Kevin's been looking at using redux, which You know has some facilities for that kind of stuff that can replay your state We don't have any media plans, but it's an interesting and open question Yeah, I will say that Kevin kind of alluded to in his talk that we're considering ways You know, we're not really at prepared to say exactly what we're ready to build right now but we'd like to improve the overall developer sort of end-to-end experience and You know Kevin demonstrated using redux as a good way to do that and a tool like that works Well with something like hot module reloading and you know, there's something that I think we can explore Without can really committing to anything. I mean frankly I think we'd love to be able to build something that community can also contribute to you and and you know If that's a useful feature, that's great. Thanks All right, so we've another question about tools Which is what are the plans for all the various tools that polymer tools team builds when it comes to polymer 3? So a polymer CLI what's gonna happen with polymer bundler will it be replaced by webpack plug-in? Etc. Yeah, so there's a little bit to be figured out there But I'm kind of on the mind that we have two different paths to explore here Probably at the same time one is we have existing customers who are using the CLI and her Successful and happy and we want to keep them moving forward into this world without them having to ditch our tool chain So we have been adding support for NPM. We now have an NPM flag on the CLI and poly serve and WCT And we'll probably try to add support for modules as an incremental choice that you can add into your project as you go But we also want to enable people to like not have to use our tools Because we're so different from everyone else to be able to use roll-up and webpack And all these other options that everybody else has out there So we're definitely gonna take care of you and keep our tools moving forward It could be we're look at opportunities to replace some of our custom stuff with something that the community already has So we have to do less work But we're also going to explore the idea of you know If you have an existing project and you want to sprinkle polymer in you can do that without having to buy into our tool chain Make Rob happy With Polymer 3 and the transition to NPM. What was the kind of reasoning for Choosing to go with yarn and a flat dependency tree as opposed to using NPM and peer dependencies It's getting all the questions here happy to do it though. It's really the platform again a Lot of people Kind of think they've been using JavaScript modules Because there's been support in all the compilers That will will turn yes, six syntax into common JS or something But it turns out that the web compatible modules are a little more restrictive than what people have been using say in the Node or like webpack ecosystems and they require that you import by path Just like HTML imports and so HTML imports and Bauer worked well together because Bauer installed flat And then you can import polymer by importing dot dot slash polymer, right? So we have that same situation on NPM JavaScript modules require importing by paths if you want to import from another package You do import dot dot slash and that package name and that kind of forces us to have packages in a flat layout And yarn does that NPM currently doesn't I hope they consider adding that feature and then we'll be able to use the NPM client as well Yeah, I think there's there's really two reasons number one is you know custom elements Have a you know when you define a custom element. Nope, that's used can't use it again So that doesn't really work well if you have to be careful about that So loading a dependency twice is bad for that the other reason is loading dependency twice is horrible for Performance, so you know relying on a tool to make your page completely not suck and load the same code over and over and over again Is not ideal. I think When we install flat we ensure we don't do that Thank you. I think actually it is worth pointing this out a little bit. So It's also important to understand that your client side dependencies You definitely want those to be flattened as Steve was saying you don't want to ship 10 versions of the same thing to the client That sucks, right? Your dev dependencies all your build tools and things like that Those can actually still have like the nested node install And so there may be a little like fine-tuning that we need to do here to make sure that like it's very easy and ergonomic For developers to sort of like be like, okay, cool. I flattened my client side stuff and I one of the things you tried out was Different yeah So like these are my client side dependencies These are my like normal nodes or build or whatever dependencies Flat and one don't flat on the other and we still have a lot of work to give everyone templates that they can start from that are Set up to serve and build in the proper way. So we'll we'll be coming out with that in the coming weeks Yeah, but like just personally like I even though there is a little bit of work and overhead involved sometimes in flattening your client side dependencies The flip side of that is, you know, I think with webpack you have What is it the common chunks plug-in or something like that? So it's like I mean you're you can either de-dupe this at install time and handle it yourself that way And at least then you've got like a gooey kind of walking you through it Or you've got to like put this whole thing into your build process to de-dupe your packages so you're doing it one way or the other usually and Personally, it seems like doing it as early as possible in the install process is a good thing Let's jump to a question from Twitter Which is what about accessibility? So how does polymer work with voiceover and screen readers in general? Yeah So how does polymer work with accessibility I mean generally like fine I can't think of anything that is like, you know painfully broken the biggest issue is going to be around Shadow DOM so Most of the stuff that we do in accessibility to build Relationships so you say for instance you have attributes like are you labeled by and things like that where you actually say This element is labeled by that other element and you use an ID reference to that other element So literally like CSS ID reference, right? The challenge there is with Shadow DOM right you're creating this little scoping bubble Your IDs are scoped to that bubble and so you can easily end up in situations where you've got something in the Shadow DOM that you Would really like to label or you'd really like to refer to you with like aria controls or something like that and You're unable to do it. You just like can't get down there So there are a few things that we're working on to fix this that aren't actually like polymer specific But are just like web platform stuff So probably the most important is the new accessibility object model which you can find that on github.com slash Wicg slash a om am and this is basically like adding a programmatic API to accessibility and making it a lot easier for you to Imperatively like so, you know like set up relationships in JavaScript between elements and say like hey I've got this element here and I have access to its accessible node property And I can actually then give it a an actual like node reference in JavaScript Then I could you know hop through shadow boundaries and things like that That's probably the most important thing that I think we're doing to fix this and that that'll that'll help web components Obviously, but it'll also just make accessibility more broadly across the platform easier for developers. I think Let's go to a live question since when I started using polymer was Because I could just serve up data on a flat But just in gen X or a static server But most of the examples I've seen now or over the past year have always been through a dynamic node server or something like that. So I just want to make sure that you know One of the that's gonna stay in polymer that you can just serve from a static server all of the things that I build are always from a static server. That's like a moniker rule so They always work the only time reduce extra things is to bundle But there is no requirement is just Inboards and that's one of the reasons why modules require paths is because the browser needs to figure out The exact URL to load and then with yarn flat it'll be able to find it and load it Yeah, I mean the one thing I would add is Because of the way that custom elements version 1 is defined because of the current existing browser support and because of our Desire always to ship really optimized sets of code to the browser We do sometimes what's called differential serving to serve just the right bundle, you know, like for IE 11 We can't serve ES6. This is why we will sort of provide tools to let you serve the optimized thing Hopefully that's gonna go away soon The next year or two I'm really happy with all the tooling and documentation you guys have provided to Help move users from polymer one then to polymer two and now to polymer three It's like, you know really made it as turnkey as possible for folks like myself But my question is, you know, there's still obviously a lot of projects in code out there with polymer one still And using the zero dot X web components polyfill What do you guys have in terms of a long-term support strategy or end-of-life plan for? Polymer one and you know obviously eventually polymer two and those zero dot X poly fills as well Can you repeat the question for anyone watching on my video or live stream? Sorry sure. I mean Taylor Yes, so question was around end-of-life support and just general kind of long-term support for a polymer one and poly fills yeah, I mean so Basically, there's the native features that we rely on is one part of the question and you know, this is sort of Web standard practice, right? Those will not be removed for centuries probably until like that We actually do crazy stuff like, you know, how many people use these things But so those won't go away really and you know polymer we are supporting hybrid mode. So I think I Can't say exactly, but we intend essentially to kind of adopt a similar policy where as long as there's need will continue to support it And unless Taylor tells me differently Does that also include the v0 shadow dom implementation in Chrome? That's a good question. Yeah. Yeah, I mean, this is really what I'm speaking to you I mean the policy bit basically one of the reasons the web is great is because it really doesn't break and you know Chrome kind of leapt ahead of everybody with v0 shadow root and You know, this is going to be available as long as people are using it The way Features are removed from Chrome is that we keep tracking them? We have a percentage of pages that are using it. You can see that as well Sorry, yeah public information from status.com. Yeah And they have to drop below like a very small threshold like zero point zero three percent Of all pages are using this feature and if it drops below that then You're confident enough that it's basically just like old pages or unused pages or just we're sorry. We're gonna break you So I don't think we're nearly close enough and we're not there's no chance. We're gonna go there in the next year one thing that is happening soon that's gonna make Monica excited, but If anybody was here still on like 0.5 or anything and you're using the deep selector in CSS or the shadow selector Yeah, that's who those are going away very soon in Chrome and that will change the styling of your page So we've been trying to urge people to get up to Polymer 1.0 and stop using that for the last couple years Hopefully you have they're basically getting replaced by the descendant selector So like it's not gonna make your style not get applied but deep is just like it becomes a descendant selector So it's not gonna go into shadow roots or anything like that. Yeah There's been a little little warning in the console for quite a while, but it's very it's weird I have this like innate reaction now. I see the warnings. I'm just like clear Like if you're like me like you gotta fix your stuff. Yeah, definitely going to break. It's definitely getting removed Yeah, luckily. I've seen some sites that I'd hopefully they've moved. It doesn't break them that bad All right, let's jump to another Twitter one How optimistic are you and what are your feelings about other browsers agreeing to native element extension? And what can we as the developer community do to like help? Is equals yeah I will take this one my cross to bear Is equals is a great idea where the the idea of extending a custom element or the idea of adding your own set of functions and behaviors and prototypes to a native element this idea very clear here is agreed on by all Browsers the implementation of this idea is not agreed on by all browsers So the current is equals implementation in particular is not is very content like contentess and is not agreed on by all browsers and this means that We would like to we will continue fighting for something like is equals It will probably not be as equals in the state that it is Hilariously enough as equals is a likely contender one letter off But basically there is a need for extending custom elements. There are things that you cannot do unless you are a native element For example submit forms so in parallel with us fighting for this API to actually work We're also fighting for browsers to actually know just fix the form element form elements should look at custom elements There's no reason why the form developed 20 years ago should not look at something developed now. So It's a hard question. We're working on it. We're trying very hard to write specs. Ritza. It's an ongoing battle I'm not I don't know if this is entirely accurate, but I'm gonna say it on stage as if it is If you add a shadow root to an element that you've type extended like so Let's one of the one of things is like oh select right select is amazing and it has all this built-in accessibility and Presentation to it and my understanding is if I extend that and then add my own shadow root like all of that goes away And so it doesn't even work. It's like let you do it. Yeah, okay Just dies so like yeah, so there's like a lot of things that you might Like think that you want to do with is equals that actually just aren't going to quite work And so then it does become a bit of like a weird API where it's like you get some stuff But not all the stuff and the browser is not very good at like pinky swing like I pinky swear I'm not going to do bad things because I've read there's like these 20 Things that I can add a shadow root to but not these other four or something. Yeah, so I definitely think personally Yeah, I want just them to sort of like expose the primitives or fix the elements I want like the label element and the form element and all these things just either just work with custom elements Or let me sort of imbue my custom element with Formness and label nests and label ability and things like that in one day one day. This will happen. Yeah promise I'll just add personally. I'm not personally. I'm not very optimistic just speaking for myself only Again, because I think this concern for example that Rob brought up around shadow roots is a really good idea of the Larger problem, you know in v0 custom elements, you could extend input But you couldn't add a shadow root to it So what you could actually do it was kind of limited why because it had this native legacy Implementation and this is really the larger concern is that these native elements have this long history You know C++ plus plus implementation and crazy specs before custom elements was even a thing So the long-term fix is to expose those APIs and make those available to any custom element Just wants them and maybe there's a shorter term fix It probably isn't gonna be is it may be some crazy thing And you know, we know that people really want this and so we're definitely kind of got our fingers on the pulse of that and And we'll be pushing for something because I think you know the exposing the guts of input so that you could do it All yourself is probably a 25 year project Yes, so, you know, we probably need something in the meantime and if is is objectionable You know, we'll be pushing and sort of pushing people for some better solution And I'll give a little shout out to Marika's talk earlier today. All of these discussions are not in some secret back room These are all happening out in the open on public mailing lists So you have good use cases and on github beyond github So you're really passionate about it Please go and like tell everybody you're very upset that is equals is not a thing because Specifically W3C or github.com W3C slash web components.  All right. Let's go to a live question. Thanks. So my question was in fact related to the previous one But first I'd like to thank you for letting us use the bleeding edge of this pack right now before it was You know recommended and given the the is attributes is gone What's the the migration path that you would propose to the currently implemented? Elements that they're using is attribute especially those that are using specific parsing context like templates script table tr and stuff like that Wait, do you want to repeat the question? I think we're getting it from the big So think that I did with form for examples of iron form was an extension of form you convert it into a wrapper and then you Put content in it. So basically all of the Extra is equals things that you were doing you put on a wrapper element, but that for tr is kind of Yeah, tables are tricky tables are tricky. Well, remember that you can style. Yeah You can style things as tables with CSS so you could move to yeah, but you got a style thing like a template What's right template time Yeah, you cannot so yeah, I mean what what Palmer did. I mean, yeah, it's not super pretty We're not gonna lie But this sort of decoration pattern is essentially this is what Palmer did with like we used to in polymer 1.0 We had template is Dom repeat we for sort of compatibility We did some magic where we sort of made that work in polymer templates in 2.0 But the way that we actually make that work is we have a DOM dash repeat element Expects a template inside of it and it works that way. This is a general pattern that's gonna basically work again It's not super pretty, but it's what you can do Thanks, thank you for this. It was great. Yeah, actually you need to bring them back That's just on the polymer project website There's still a mention about an upgrade tools to 2.0 from 1x any update on this On the upgrade tools. Yeah, there was a mention about it was from 1.0 to 2. Yeah So, you know last year we said we're we're gonna make this upgrade tool and then This is kind of my bad a little bit It seemed like most of the feedback was like, oh, it's really easy to upgrade and the things that were tricky to upgrade We're actually gonna be impossible for a tool to automate things like converting a content Tag to a slot tag where the content had actually had a selector. There's no way We can just automatically determine what the slot name should be everywhere and apply it So it seemed kind of to us for a while that everybody The we were hearing from was kind of satisfied with how easy it was to upgrade and then it seemed like more recently We heard an uptick in number of people who were asking about about the tool and wanted it So also recently we started converting all the internal Elements inside Google from one to two and a Peter an engineer on the tools team has been working on the tool Quite recently. It's not very clean code. We've just been like hacking at it to do whatever upgrade we could And so we're planning on releasing the version that we have for internal External pretty soon. Okay, then do you have a suggestion to lint old element to find? You know quickly what has to be changed to zero right rather you know going to each element and you know searching for specific things But you know something too late, so it's highlighted a Way to highlight in your code, you know what to be changed like Yeah, we can certainly if there's cases where we we can't determine what it should be We can output a message that says, you know, this needs to be updated by you Okay, thanks. All right, we're running long time. Let's do one more live question. Hello My question is Do we have any Tutorials or videos for the making something like neon animation because we really loved in the love the neon animations in the polymer one I think the question about neon animations in its future, right? So is it is is it if it is we have a blog post that Elliot wrote you probably saw one stage before On the polymer summit site that kind of explains what happened to neon and animations and what's going to happen to neon animation Yeah, so yeah, my question is like Is there any some new videos or tutorials like neon animation because in the version to it is deprecated, right? Will there be replacements for neon animation? Yeah, I don't think so the problem with you on animation is that it's fundamentally a Flawed concept that we thought was going to work in dozen like it doesn't actually add anything in the current state that it is amazing to the web animations polyfill API and We had a team the team that actually works on the web animations API Look at making a new and better neon animation and what it ended up being was the web animations API So it turns out just like a declarative rappel would be amazing, but it just doesn't really work well, and If you want it just a jar of spiders Yeah, I'll also plug Valger and another engineer on the team made a code lab for performant to expand collapse animations and Just by going through that you can see you know something Pretty typical that anybody would want to build and how he does it and then you can learn yourself and maybe apply that to some other projects Thank you Let's close out with a question that I love which is What would each of you see as your number one Request or a current hole in the web platform a feature or some change that you would like to see made to the web platform to make it better There's two I'll go one and hope somebody else does the other one I think right now for web components We're looking pretty good with the native implementations at least in chrome and safari The one piece that seems to be left to make everything much better is the theming support in CSS for shadow dom So that's the part and theme CSS spec it just got promoted to working draft or editor's draft. That was a good one. I want to take the other one I know it's fine. Okay So yeah, that would just help that would just help everybody's theme across their shadow roots That was mine For better or for worse for some reason I care about inputs and forms despite never actually Using a form on anything that I've ever made so what I would like to see in the web platform is Input being less opinionated and forms being less opinion native elements being less opinionated about their tag name So form caring about more than just input and you know input type equals color actually working on all of the browser is because it's Not yet Yeah form an input I would like form an input to succeed at their job Yeah, so mine. It's kind of related to that like there's a lot of things that native elements can do There's like a lot of built-in magic I've been working on accessibility a lot the how-to components that we all worked on Accessibility was like the primary driver behind that work We were like how do we build tabs and trees as custom elements and make sure they are like really like Robust and accessible right fully keyboard accessible doing all the right things with aria everything we can possibly try and Then there's still places where we come up short because there's some stuff that we just can't do in the browser today The label element and things like that you click on a label and it focuses your control That's a cool magical behavior that like Just there's no easy way to get that you're gonna have to make your own custom element label now And then you end up just reinventing like all of html to get all these little features So I'm really excited about a on and other you know other stuff like Being able to hook into forms. I think these are probably some of the most common things folks have brought up And we've kind of been pushing for them for years. I actually think people are listening a little bit more right now Definitely both forms. I think forms are actually a good Yeah, good hope we're like finally getting through just some folks there So yeah, that's really what I want I want I want to make sure the future of html is accessible and that's easy for developers to do that to do the right things And it's not like this arcane art to make something accessible I think that The biggest frustration I have is when there's like a really cool feature And not every browser has it of course, you know trying to bring down polyfills And I think a way that that could happen is Oddly enough through payments and commerce on the web because the more we allow users to buy things and spend money there The more companies pay attention large companies that have tons and tons of sites and have tons and tons of tons of developers And so we can start doing more stuff with payments I know that Chrome payments just came to desktop We'll start attracting a lot more attention and then hopefully companies will start putting a lot more attention into their browsers or paying attention in standards meetings and stuff Yeah, people on the team know that the thing I care about sort of the most is performance And I think I've said this before but you know I think with custom elements we have an opportunity to add things to the browser that makes it faster Makes us able to do things custom in custom ways faster I care about this for a couple reasons Obviously, I want things to be faster so that we can all make stuff that's awesome and better for users but there's another reason and You heard yesterday and Alex Russell's talk. He's talked a lot about mobile CPUs Now he only talked about that for like five minutes but he works in the same office as us and We have to hear him talk about that for like hours and hours and hours and hours and I would like to have things faster. So I don't have to hear that venting Much anymore Alex, he's joking. We love to talk to us All right, great So Matt's gonna come on in a second with some closing remarks and some very important announcements So please be sure to stick around but thank you so much to the panel


Broadly there are two types of rigidity posters, de-quarticate rigidity poster and de-celebrate rigidity poster. Let's try to understand the mechanism of how they occur. For that we need to understand about the muscle tone and what are the various tracks which influence the muscle tone. So we have to see it in both the flexor muscle and extensor muscle. So suppose this is a flexor muscle and this is an extensor muscle and we know that skeletal muscle has intrafusal fibres and extrafusal fibres where extrafusal fibres are supplied by alpha motor neuron and they contract when there is a stimulation of alpha motor neuron. On the other hand intrafusal muscle fibre is a sensory, right? So there are sensory nerve endings from intrafusal muscle fibres which reach to the spinal cord and this has connections with the alpha motor neuron. Then the ends of the intrafusal muscle fibre are supplied by gamma motor neuron and these ends are actually contractiles. So when there is excitation of the gamma motor neuron there is contraction at the ends of the intrafusal fibres and this intrafusal fibres stretches and because of this stretching there is increased impulses from these ascending fibres that is 1, and 2 which go and excite the alpha motor neuron. So this is the basic stretch reflex actually the connection between the sensory fibres and with that of the alpha motor neuron. Now these motor neurons that is the alpha motor neuron and gamma motor neurons are influenced from neurons from top and there are various influences there is influence from the cortex then from the midbrain where red nucleus is there from there then pons and medulla actually there is a reticular formation in pons and medulla and from there reticulo spinal tract arises which influences these neurons. So let us see some details about these. The flexor muscles and the extensor muscles receive different influences. Flexor muscles alpha motor neuron receives excitatory influence from cortex. So that means from the cortex the fibres which are coming they are exciting the alpha motor neuron of the flexor muscle also via red nucleus. The cortex influences these flexor muscles actually from red nucleus alpha motor neuron receives excitatory influence and this red nucleus receives inhibitory influence from the cortex. So this fibres which are coming from the red nucleus of the midbrain to the flexor muscle alpha motor neuron this is the rubro spinal tract and one thing you remember that this tract only descends up to thoracic spinal level. So it is not going to influence the flexor muscles which are coming below this level. Okay now let's see the influences on the extensor muscle. Extensor alpha motor neuron receives inhibitory influence from red nucleus. So are you getting it that the influences from the cortex and from the red nucleus basically are exciting the flexor alpha motor neuron that is causing its contraction. However they are inhibiting the extensor alpha motor neuron. Anyways now the extensive gamma motor neuron also have lots of influences. The gamma motor neuron supplying the extensor muscles receives excitatory influence from the pons reticular formation but inhibitory influence from medullary reticular formation. Okay now both these tracts that is the medullary reticular spinal tract and pontine reticular spinal tract both receive excitatory influences from cortex. So in summary corticospinal tract and rubro spinal tract from red nucleus they excite the flexor alpha motor neuron and then there is reticular spinal tract from pons and medulla. From pons reticular spinal tract is exciting the gamma motor neuron and from medulla there is inhibitory influence on the gamma motor neuron. So reticular spinal tract works by acting on gamma motor neuron. Fine now let's see the two types of rigidity posters which we spoke about in the beginning. So first one is the de-cerebrate rigidity. In de-cerebrate rigidity what is happening that these tracts that is the cortical influence on the alpha motor neuron and the red nucleus influence on the alpha motor neuron both are done away with this. So basically the lesion is somewhere here below the red nucleus and that is basically a mid-collicular lesion. So in this lesion all influences from cortex and midbrain are lost. So you can predict what is going to happen, where it will be affected. Simple you see excitatory influence to alpha motor neuron of flexor muscles is lost. And what will be more active it is the influence on the gamma motor neuron coming from pons and medulla. But we said that the influence from pons and medulla one is excitatory and the one is inhibitory. Then that means overall there is no influence no that doesn't happen like that. Because actually this excitatory pathway from pons reticular formation, pontine reticulo spinal tract dominates because the sensory signals from the intrafusal fibers which are coming facilitate this pathway. So in simple terms due to increased activity of pontine reticulo spinal tract which is basically having excitatory influence on gamma motor neuron the stretch reflex becomes more sensitive. As I told earlier that this is causing the gamma motor neuron is stretching the intrafusal fiber and this will further increase the afference activity coming from the intrafusal fiber and hence it is going to activate the alpha motor neuron. So if alpha motor neuron is activated extensor muscle is going to contract. So what is it? We are getting a posture in which the extensor muscles will be in contracted state but we have seen how flexor muscles will not be in contracted state because the influence from the red nucleus and from the cortex is missing. So in de-celebrate rigidity what we see is that both lower extremities and upper extremities are extended. Lower extremities are extended with toes pointed inward and upper extremities are also extended and neck and head are also extended. Now let's see the other rigidity that is the de-corticate rigidity. Now as the term indicates the de-corticate rigidity basically the influence from the cortex only is lost so the lesion is above the red nucleus. So only the cortex influence is lost and red nucleus influence is still active. So what will happen? See obviously the extensor muscles will be similar like that of de-celebrate rigidity because the reason is same the afferent impulses coming from the extensor muscles are activating the quantine reticular formation and because of that there is extension of the limbs. However this red nucleus excitatory influence to flexor muscles is intact and remember as I told you earlier that this influence is up till only to the thoracic spinal level. So if we consider the muscles right then this influence is only on upper limb muscles. So in de-corticate rigidity we get extensor hyperactivity in lower limbs but flexion of upper extremities at the level of the elbow. If you like the video do press the like button and do share the video with others and don't forget to subscribe to the channel Physiology Open. Thank you.


Znamo se razvršenje, da je dobro vse dobro odličenje klas. Vsih zelo je več nekaj zelo za vse. Znamo, da se pričoče... Zato, če je in izvaj, in nekaj, nekaj ne zelo. V tudi, tudi, tudi lektor. Če je nekaj nekaj klas, pa je stavnja klaso, da je zelo, da je Kaminu, da je Leldi, če je zelo. Zelo, da se vse, dont we are going to present, this is something of the regularity of minimal services. And actually of what Camilo introduced, this are area minimizing, OK? Camilo sho you, in theory of internal currents and some of the issues that you see in the regularity for Chodimation 1 currents. So, especially he gave a proof of a partial case of the Georgian decay lemma, which is at the base of the study of the regularity for collimation 1 current. In this week we will see something related to higher collimation, so this one will be the focus of my course. So, understand some of the issues which are involved in the study of regular points and singular points in higher conimation, minimal surfaces. And the plan of the lecture is, in some sense, to focus on some of these issues, which are kind of basic, in my opinion, but in this theory, but in the simplest case we can admit. So now this one will be clear in a while, so we will try to avoid all the technicalities to focus on the geometrical and analytical problems which are behind. And maybe the best things to do is to start with an example and always to keep an example in mind, which I saw Camillo already introduce, and that's the example. So that's kind of introduction via examples. Camillo already introduced a class of minimal currents in higher conimation, which are the complex varieties in CN or in a scalar manifold. So since it's just an example, let me just introduce the most basic one, which is a complex curve in C2. So the one which Camillo introduced, I think, was ZW in C2, such that Z2 is equal to W3. So let's understand a bit the geometry of this complex curve. So my favorite picture is try to write this as a graph, which is clearly impossible, but gives kind of the ease to the geometry of these objects. And if we write kind of naively, we have that Z amount is one of the two square roots of W to the cube. So let me write that's not correct, but it's plus or minus these two square roots. In some sense, if here I put a two-dimensional plane, so Z in C, which is a real-dimensional plane, and here I put W in C. So what I see is plus or minus, so ZT is plus or minus W to the 3 half. So something which looks like this. So this one is V. Now what is V actually? If you see here, it seems kind of disconnected, but it's just a two-dimensional projection or of a four-dimensional space. So this V is living in C2, which is a real-dimensional form, which has four real dimensions. So what is actually V? V, if you see, it's a very nice topological disc. It's the image of the flat disk in R2 via a smooth parametrization. So V is equal to U of B1 for B1, the unit bowl in R2, and U goes from B1 to R4. In the following, I will always identify this introduction, and see with R2. I mean, we can also give a parametrization for such map. So in polar coordinates, a parametrization is the following. So rho cos theta, rho sin theta, rho 3 half cos 3 theta, rho 3 half sin 3 theta. So it's the image under a continuous map of a disk. So actually it's a very nice surface, much more regular than a general current. And if you see, this parametrization is actually very smooth. It's an embedding everywhere except at the origin, where it is just an image. So this one is just to put all this information together. It is an example of a minimal current in R4. So it's a two-dimensional minimal current in R4, which is regular everywhere except in the origin, where it is just immersed. And there's a kind of double point there. So what we will try to analyze in this week is the behavior of these singular points here, which are called bridge points. So in some sense the aim of this, of what I'm going to present, is to find some analytical way to understand in the presence of this such behavior. And this surface here is a very clear example of why the techniques and the tools in codemation 1 cannot work in higher codemation. Because, as Camino showed you, the main parameters in codemation 1 for the regularity was the excess, which is an integral measure of the oscillation of the tangent plane. And here the tangent plane is continuous there. So remember, this one is an immersion. It's a C1 immersion. So in some sense looking at the excess in this case would not detect the regular point. So that's just a remark. So the excess of this current in Br, around the origin is going to zero. So at a certain point becomes less than the constant in Allah's theorem, but nevertheless the surface is not regular. So just looking at the excess is not enough to understand the regular, the singular points. And indeed what we are going to see is something which is completely different because I'm out from codemation 1. A different even in the results. So maybe I give you immediately the final results, which is something I mean, we won't see in this course. It will be too long. But it gives, in some sense, the idea of why this theory is so different from the one in codemation 1. And the results, I don't know if Camillo mentioned to you. In higher codemation, so in codemation 1, so let's say, so which is the case of hypersurface, so we have that every minimizing current is regular. So regular means that it lives on a smooth manifold and is given on integration by integration over this smooth manifold. It's regular in the interior except for a set of at most m minus 7 dimension, so out of dimension, where m is the dimension of the current. And this one is the most basic result in codemation 1. Actually much more is known, but just to point the difference, we just talk about out of dimension. Which I mean, for those of you who do not know what is out of dimension, just keep dimension. It's a very reasonable notion of dimension that you may introduce in this problem in geometric analysis. This one is a result of a very long analysis and effort, which starts with the Georgian and then was completed just in the 70s by a great federal, other than many other authors, Simon's and so on. But what is striking is the final result. If now I would like to write the final result in either codemation, you see it changes completely. So in either codemation, the singular set may exist. So I mean, the question is where this magic number is coming out. It doesn't mean for me it's I think as magic as for most of the people. So if you see the computation, it comes from an analysis of the stability of cones and then you see or geometrically, you see as a possibility to foliate a space with a foliation, which is singular in one of the leaves. But why this comes first in this dimension I cannot say. Honestly. But I didn't say what is my degree of optimism because now I knew before the deal. So in either codemation, instead, the singular set may have at most, so is at most of dimension. So I was warned not to write down here, so just change. Report is at most minus two, where m is always the dimension of our current. So you see what changed is not the techniques and the way now I show you how to approach the problem, but the result itself is different. And this result is optimal. So we have shown already a curve, which is a two dimensional object, so m that is equal to two, where we have a singular point and the singular point is dimension zero. So this means that it can happen to have a minimal current with a singular set, m minus two as dimension. So it's not that we cannot prove more than this, but it's that this result is sharp. And this result, this theorem is due to angle, so in the x. So now what we will do in this class is in some sense to understand some aspects of this theorem. Revisit it with some modern terminology and modern concepts. So this one is somehow the starting point for our analysis. And again I think the best way to understand where to start is looking at the examples I gave you. So the examples was the one of the complex curve B, which is roughly speaking z equal plus or minus w3r. But somehow correctly is the one given by the parametrization. I wrote you before. So, and why I would like to start from this picture, because it gives somehow the idea of what we are going to do. And the point is in starting minimal surface, also in this very weak context of minimal currents, all the effort is done somehow to reduce to the case to a nonparametric problem. So the case where we can use an equation, the minimal surface equation or the linearized equation which is the Laplace equation which is what Camilo showed you for Allard's theorem that once you reduce to a graph, to ellipsis graph, then you may prove the decay of the excess and then you transport this to the case of minimal currents. Now the same would like to do here. So we would like to write a nonparametric problem for these objects. But now you see, so from the picture it's not so clear, but I mean you may realize by yourself that you can write this surface as a graph of a function. So morally the tangent plane with respect to which we should parameterize this one is the tangent plane of the parametrization of the immersion which is this horizontal plane here. So somehow the first claim is that this horizontal plane is the preferred one to write a parametrization. But around this point there is no way to write as a graph of a function. And the reason is because of this branch point. So if you start from say now let me write this at the plane W here. The W plane. So if you start from a determination of the square root at E1 say and then you follow a circle around the origin, you will end up somehow on the other determination of the square root. So there is somehow no way to write this as a graph as the union of two continuous graphs over the W plane. And that's because of the nature of the branch points which is exactly defined by these properties that when you follow a continuous path around the point you will end up with another determination of your leaves of your coverings. And that's what prevents to have a representation via a graph. Unless I mean we are allowed for a singularity here, but we would like to solve an equation which gives regularity, so we cannot have something which is discontinuous. So somehow starting from the beginning it's kind of clear that we cannot use a very classical non parametric theory for treating this problem. And here comes the first main idea of Andre, which is okay. This one is not the graph of a function, but above every point in the W space, we have two values of the z. And always two, also in the origin because this one is a single point but it's kind of approximated by by two points. And if you see the density in terms of currents this one is a double point. So the density there is two. We cannot treat this as a graph of a single value function. Let's treat it as a graph of a multiple value function. So in this special case the function is so we can write kind of formally. So to each W we associate two values which are in so let's write, I mean kind of informally, but it's clear which are W to the three half and minus W to the three half. So I have a map from a unit disk, say in R2, taking two values in R2. So which is this set here which is a space that in a while I will introduce and we call atomic measure when mass two of R2. And now what is one of the main points here? One of the main issue in this parameterization that when I give you a determination of the square root of a complex number I cannot tell you in a consistent way which square root I am taking. Because as I showed you once you follow any continuous path on the plane in the branch point, we change this determination of the square root. So in this representation what is important is that here I do not use this bracket so it's not an order pair. I'm not saying what is the first value what is the second value. I just give you the set of values and that's as a whole, it's my value of my function. So this space here is not R2 squared. But it's kind of new space of points which is the space of unordered points. So that's kind of the beginning of the analysis. So let's start from introducing this space. Both times the same but now it's clear from the definition and this one is the space of in this example R2 we will always consider Q point which was traditionally introduced by and let's put a sign here that means that these Q points are not different Q points so in the way to formalize this concept is to take so there will be the Q points in Rn and I consider this as the atomic measures of mass Q in Rn. So positive atomic measure of mass Q in Rn. So I write the sum e from 1 to 2 of the Dirac delta in Pi so this one is a Dirac delta such that Pi are points of Rn. And then you see these points are not necessarily different so for example a bridge point could be two times zero. So our values will be always a measure which is atomic so leaving on delta and mass Q so positive with mass Q. So this one belongs to aQ of Rn. And just to understand a bit this space that's just a very nice formalism because we can think a bit about summing measures or stuff like this but another point of view to see this space is actually the combinatorics one I was kind of suggesting here. It's true that it's a collection of Q points in Rn. So this space you may also see as Q copies of Rn. But then you have to take care of the order of the points. So you have to take a quotient where the equivalence relation of permutation. So P1 P2 is the same as P1. And this one gives you a bit the idea of the geometry of this space. So it looks like a nuclear space because it's actually R to the NQ but then it's you took the portion with this group of permutation. So you create some corners and some angles in this space. So the space in the end will turn out to be not a smooth space naifli. So let's say the first structure we can give in this space is the one of a metric space. So let me introduce the metric for this space. So here we have a lot of freedom to design which metric but it will be clear in a while why this one is kind of the correct one. Traditionally it's written like G for the metric so this one is really the distance between two Q points T1, T2 and imagine T1 is the sum of the data in PI and T2 is the sum of the data in SI. So this one is the minimum among all the permutation of the square root of the sum always 1 to Q of the Euclidean distance I minus S sigma I squared. So just to understand this with the picture we have two sets of Q points so let's take the case of exactly I don't know two points so T1 and T2 and we consider the pairing of one point with the others. We have once this and then we have another one. No, this is the only one I shouldn't know it's this and this. Then we see which one realize the minimum of this distance here and this one will define the optimal pairing between these points and last week you had a week of optimal transportation so I think for you it's kind of now very clear that it's actually the vastest time to distance in the space of measures. In this very special case of atomic measures so that's nothing else than the objects we have already seen I think already in this course and now it's very simple but this one is I will leave you as an exercise that this space of Q points with this metric and by the way I posed some notes for the class so some of the material you find here and also some of the exercise with the ins also to solution are in the notes if you need to consult but not this one this one is very simple OK, so now we have our space of points let's define what is a function to this space of points so nothing easier than this space and we just consider functions between two metric spaces so let me I should never erase the picture of the complex variety that should stay there with us so now which kind of function we would like to look in this problem so our function from a flat space a is this so these are two valued functions which are just map U from say an open domain so open bounded and let's say also smooth that's not the point in a flat space and M is always the dimension of my object of my current to this space of two points in Rn so which may be one or greater than one in important cases is when n is greater than one so in this picture it's like that here we have Rm as the horizontal space and the vertical space will be our Rn and we consider two points in this Rn so that's the splitting of our and now this one is of course a metric space here I introduce a metric and I can talk about continuous function measurable function the same as application between metric spaces so this one is kind of clear the next step is to try to define a derivative for such a function because remember our aim in the end will be in some sense to analyze the minimizers of the area of the mass of these currents in this graph it's expressed by the integration of a first derivative so that's so the area of the graph of u is the integral of the Jacobian of u a Jacobian is nothing else than the square root of the determinant of du transpose du so somehow we need to have a notion of of derivatives there and now this space is perfectly nice it's Rm but this one is a metric space so the point is trying to define for this class of functions a notion of derivatives there or or weak derivative so what we introduce now are generalized sobolev functions ok let me say there are many different ways to introduce sobolev functions between metric spaces and here actually it's also easier because this one is a nuclear space and but the way I find more convenient is a way which is going back to to Ambrosio in the 90s which is looking at composition with the Lipschitz functions so I say that the function is sobolev and the Lipschitz function is a sobolev function so let me give a formal definition so we say that u from omega into this space of two points and when I don't indicate the dimension is kind of always a medend there is let's say w1p if there exists a function phi in lp of omega and this one now is the usual lp space so this one is a function from omega into r and actually I mean we can take positive that there exists phi in lp such that the following points and we need two conditions here so the first one is that every composition is a Lipschitz so instead of taking generic Lipschitz functions I will take composition with a distance function so I say for every t in my space of two points the map which now it's a real valued map so it goes from omega into r and each x I associate the distance of u of x which is now a number so this map here it's now actually a real valued map and I ask that for every t this map is sublev w1p so meaning lp and the generalize generalized gradient but then what is important is that all this composition needs to have a common dominant for the gradient so then which is this function phi here I ask that the norm of the width derivative of this composition so for every t the norm of this width derivative is point wise dominate by phi a point wise means of course the width derivative almost everywhere in omega no that's point wise so for almost every x and it's a finite value and has to be controlled by this function phi and also phi is defined almost everywhere because it's just an lp function so it's really point wise almost every x say in omega that's what we ask now this one is an exercise which I put in the notes so that's somehow the most in this space but it's like asking this property for every Lipschitz composition and that's what is important exercise belongs to w1p omega aq if and only if composition u belongs w1p omega and the derivative f composition u is less or equal than the Lipschitz constant of f times phi almost everywhere and that needs to be true for every f Lipschitz from aq so it's to remember it's a metric space so every Lipschitz function from aq to r that's an exercise I've been asking that property just for the distance function of the Lipschitz function a Lipschitz function it's a way to know a metric space somehow so that's why this kind of notion it's so reasonable question on this because this one is one of the key points of what I'm going to present and next one is the energy so trying to define a the area function for this kind of multiple values so in some sense we need to define so the next part of the hour we will spend to understand how to define the area function or a variant of this now we will see and the first step we do here is in some sense to immediately to simplify our object which is also what is done in the theory instead of looking at the area function we look at the linearized energy so as I wrote you so the area of the graph of a function u is given by the integral of the Jacobian and now kind of linearized this function in the first order you get actually this one is the integral of one plus du squared over two plus a reminder which is a big of du to the fourth so in some sense a very cheap way to define this energy would be just to define the modules of du and that's what we are going to do most of this hour will be to understand this object so this formula don't take too seriously that should be another u which is the one which is also the first components of the identity but it remains true that this one is the expression so now try to define du and looking at this definition here this is a very natural guess which is ok, I take the minimum phi realizing these properties and this one is my du because every gradient of this one leap sheets map is dominated by phi so this one is very reasonable but it's the wrong energy and actually the energy will be defined and will be clear at the end of this hour excited by this property here of the area function so it really needs to be the directly energy and now if you look at this energy here for vector value map this one is not the directly energy this one is the operator norm of the gradient so it's not the invert smith no, so we need to do something else and for this the expression of the energy we introduce is a sum on the partial derivative in which case the two norm coincide now we have a map like this and it's a different u you have x, x u of x so this one is I should put here say big u x u of x so that's the map and this one the derivative of this gives the one in the expansion ok so what about du so we define so before defining the modulus of du we define the partial derivative the modulus of the partial derivative of du so we take dj of u and look I write like this but this one is kind of an independent object so it's not the modulus of a derivative up to now but it's just a quantity positive quantity which will be an LP function and it's given by the optimal phi there for the partial derivative in j which I mean it's not difficult to regularize that this one is the supremum over a dense a countable dense set of of points in this space of two points of the modulus of the partial derivative of this composition so I select a countable dense set of points there I look all of this composition all the partial derivative and I take the supremum point wise almost everywhere for this quantity and then the right quantity to look du squared is exactly the sum on the partial on the on the different direction from 1 to m of this partial derivative so this one now we will see in a while why it's the right quantity for the energy and note that in general when we have a map with the values into a metric space it's very simple there are many different definitions for defining the modulus of a gradient but not for defining a point wise gradient this one is something which is general much more subtle in this case I mean we can define also a point wise gradient and now we will spend the rest of today lectures for doing this in order to understand a bit better this class of maps here and we will see that this point wise gradient will match this energy we have introduced here so up to now all the theory up to now it's possible to be done for generic metric space here so complete metric space here you may define this energy so breath class like this and from now on we exploit the structure of two points actually a bit more in details so now we try so let me see if I forgot something to give a notion of point wise differentiation so let me first say something on the representation of this multiple values function so given a function u from omega to this space of two points it's always possible to find q in terms of number different measurable functions would sum as measure gives this function u and I mean huge unique measurable so u1 uq from omega into Rn such that u of x is exactly the sum of Dirac deltas in ui of x so this one is a very a very useful way to name the values of this function there so in some sense I'm not claim anything about this ui so I'm not saying for example that when we start from a continuous function the ui are continuous that's not possible or when we start from a sublet function the ui are sublet I'm just saying that we can give a label in a consistent way which is very non unique superposition superposition of measurable functions ui but in general this one would be just measurable and the proof of this statement is by induction I gave like an exercise in the notes and there are hints for the proof of this and it's not so deep I mean it may seem strange somehow because I said we cannot order the value but when I said we cannot order in a consistent way in such a way that I mean when we have a continuous function our values are ordered in a continuous way that's impossible but in a measurable way so just giving a label which is measurable we can do and that's I mean it's always possible and we will use this writing here just as a way to write our function so nothing special just having a way so that's the only use we will make of this proposition which I mean by the way the proof is very simple by induction and it's given in the notes so you're measurable, yes so that's kind of the minimum regularity I always assume OK, what is wrong is when you substitute this measurable word with any other words then so if you have leap sheet and then you find Q leap sheet function it's impossible so obolev that's the only wording in this kind of statement so but this one is useful because I mean it allows us to use this notation and it's to introduce this notion of point wise differential so definition so we say that U is differentiable X naught in omega if L1 and I mean we see as linear map Rm to Rn so linear map and now we give the two condition for the differentiability there so the first one is that now we set a notation for the first order differential of a map so T at X0 view that's kind of the tangent map to U at X0 which is let's use now Y as a variable so this one is the sum of these deltas at Li Y minus X0 plus Ui of X0 and then here you see the first use it's just a way to label these values so this one is a sum of Q different linear maps based this affine map at Ui of X0 so once we set this then we have that the distance between U of Y and this linear map here as usual for a differential is a little of it's an infinitesim map of the distance so that's so I'm telling you what is the linear the differential of this map and I ask for the usual differentiability properties that it has to be infinitesimal with the distance to the form so this one is quite it's quite natural but there is an extra condition which makes this notion a bit more precise and an extra condition is that every time here a point with ion multiplicity also the linear map should coincide should be the same so Li is equal to Lj if Ui of X0 is equal to Uj of X0 so now let's explain this this definition with a picture so basically what we say is that a point is so let's do the first case where we have exactly the superposition of two different smooth functions so let's say Q equal to smooth function which are kind of disjoint so we have here our two functions and our two values function is the superposition of this very very smooth two function then at a point zero in the domain we may take the differential of these two so this is a fine map here and our differential for the map will be just the superposition of these two differentials and this one will be a point of differentiability for our map another case is when the function are not disjoint so always Q equal to but we have a situation like this so like a branch point for example so this and this so the map comes together and this one is our point X0 then we say that X0 is a point of differentiability if we can find a double linear map which approximate this function there why double? the linear function has to coincide so here we need to find the common tangent plane to the two which approximate to the infinitesimal of the distance our two values function so this one will be taken with a multiplicity 2 and this one is still a point of differentiability for us what is not a point of differentiability and it's kind of questionable but this just make this definition a bit more restrictive is when we have a two value function which behaves like this in principle now it's linear let's do less linear in principle here we have two very nice differential function and the superposition of these two is a good approximation for our function u but nevertheless we don't consider this point the point of differentiability because the point is the same but the linear map are different so that's what we exclude from our definition of differential point and I mean the reason why this case is excluded now it's a bit vague I cannot explain kind of completely but it comes from the fact that in some sense all the all the core of the theory is to look at the space of q points as a union of stratified space so the two the multiplicity of two points would like to see as independent from the multiplicity of one point so to do this means that they needs to have their own differential which is a multiplicity of two differential so the definition may seems a bit weird because that picture is perfectly nice in terms of surface that one is kind of an image with a self intersection of surface for the theory is more convenient to take and actually suffice to take this more restrictive definition but it's mostly used later this so at this point it's not completely justified why we consider this second condition here ok, and now we make a little break of ten minutes and we assume at 9.40 and for the second hour I mean we will see that actually this kind of point wise differential exist almost everywhere for sovereign functions and we will give another way to write the energy which will be kind of more convenient for certain computations ok, so we assume in ten minutes here ok, let's start again very very slowly ok, so now we have a notion of differentiabilities for our maps which is this one which is shown in these pictures it will be completely unuseful if we cannot say that our function is differentiable so the first thing we have to do is to prove a sort of rather marker theorems so that at least our differential almost everywhere for our functions so this one will be next step which is a generalized for the markers theorem so which at the beginning I mean we will state just for for Lipschitz function so the theorem will be we take f from omega to aq the Lipschitz and I recall you the notion of Lipschitz function between two metric spaces it's the standard one the distance is not expanded up to a factor L then so then is differentiable almost in omega so this means that for almost every x0 in omega we find such a correlation of linear maps as it is I wrote before OK and now we try to give a proof of this result so won't be a complete proof but almost complete let's say because all the ideas of the proof will be inside here so proof with more sketch and what we do to illustrate this phenomenon of the proof is to consider the special case of two values function so q equal to I don't know if I started a bit in advance so and the point is the following as I told you the idea what we to treat this multiple value function is somehow to go to go down and distinguish between multiplicity one, multiplicity two and so on in the activity with the formalism now which is completely independent from these combinatorics because we have this notion of metric space but nevertheless this one is kind of behind this argument so when q is equal to what we can do is distinguish between points of multiplicity one and multiplicity two so we call omega tilde let's say the points x in omega with the multiplicity two which are the one where the two values of our function are equal and then we look separately omega tilde and omega minus omega tilde and notice that omega tilde is a closed set because f is it's a it's continuous and this one is defined by inequality so it's very simple to see that it is a closed set ok, what happened here so outside of omega tilde we have two values which are different so let's argue with the picture so for every x naught in omega minus omega tilde f1 of x naught is different from f2 of x naught and so we see basically so that's our x naught in this case we see basically two different values for such a function now the function is continuous so now I'm not going to prove this but it's a leap sheet it's a leap sheet continuous the two values are different this implies that these two leaves of the function cannot meet in a neighborhood of this point so there exist we see two different two different leaves of my functions and now the function was globally leap sheet it's very simple to see that f1 and f2 are also leap sheets in a neighborhood of x naught so in some sense in this neighborhood we can apply the classical or the macro theorem and find differential for both functions almost everywhere I mean we cover this set so here and now it's a very simple to verify via the definition so by the very definition at the beginning this map here so i1 to 2 of dfi x naught y minus x naught plus fi x naught this one is a differential for our two values function so I say a differential but you see from the definition that when exist the differential is unique so it's the differential or the derivative so when we have these two values which are split then we argue as if we had normal normal or classical function so the only thing we need to understand is what happened in omega tilt is it clear this with this classical or the macro theorem which is said that when you have a Lipschitz function it is differentiable almost everywhere with respect to the back mesh so all the things we need now is to understand this multiplicity 2 set in omega tilt f1 is equal to f2 and now omega tilt is not the entire space but we can use theorem for Lipschitz function so for example the Kitz Brown theorem and say that f1, f2 coincide with the restriction omega tilt of a Lipschitz function ok so on this set the two values coincide the function there was was Lipschitz so in particular considering these two values kind of separately they are Lipschitz they are the same value and they coincide so with and extend them and look an extension of this Lipschitz function and now this tree I use for defining which points I look for the differentability of of this and I say that I consider points x0 in omega tilt with these two properties so the first one is that they need to be points of density 1 for omega tilt which I recall you so does it mean it means that if I look at the measure of omega tilt in smaller ball around x0 it has to cover almost of the ball so the limit are going to 0 of Br x0 intersected omega tilt renormalized by the measure of Br is going to 1 so it's covering everything around this point x0 and then since G is Lipschitz itself I look at the point of differentability for G so x0 of differentability for G and now what I claim is that for all these points x0 with these two properties actually my function f was differentiable and notice that by the lebek theorems almost every point in omega tilt there is a density in one point and this condition which is satisfied almost everywhere and now here I apply again my Rademacher theorem in the classical case and I know that G is differentiable almost everywhere so these two conditions are satisfied almost everywhere so would suffice to prove the differentability for points with these two properties it's a full measure set so that's actually a theorem which is not a trivial theorem which is based on the basic of which covering theorems which says that if you take as a measure for example the lebek measure restricted to the set omega tilda you can find a differential almost everywhere with respect to the lebek measure and this one will give you the density of the measure which is 1 on omega tilda 0 outside so you know that almost everywhere is coincide with the density which is 1 inside and 0 outside but this theorem I mean it's not trivial at all it's a very deep theorem in measure theory so now we are we are left with this task so which is to prove the differentability at this point there so let me fix some notation so the claim claim is so f is differentiable at m so let's make this claim a bit more precise the linear function to f at x0 is exactly twice the linear function associated with g so twice dg x0 y-x0 plus g of x0 of x0 so this one is my claim so let's prove this so we set r equal we consider a point y in omega and set r equal the distance between y and x0 so now always with the picture so this one is our point of our candidate point of differentiability we have our set omega tilde which is a full set around x0 so something may be missing from omega tilde but always few in few once we approach x0 and now we try to verify the definition of differentiability at this point so we consider another point y and we try to prove that the distance between the linear part and the function is infinitesimum with the distance of these two points and what we do is so we consider the bolt which is twice this b2rx0 and let's take the closed bolt and now what we do is so in some sense when y is a point in omega tilde we know what is the function there because in omega tilde g was coincided with our with our multiple values function but when y is not in omega tilde we don't know what we are doing so the best is first of all to find the projection of y onto this set omega tilde and we call this projection y star and y star is a point in our boot set omega tilde in so much that the distance between y and y star realize the one between y and the set omega tilde inside this closed bolt and now we will always verify our condition passing through this point y star so as amount here what we use is the fact that everything here is Lipchitz continuous and once our Lipchitz continuous here it's too low let me go up here so then in order to estimate the distance between f of y and the first order differential at y we can put in between our comparison point which is this y star so using the triangular inequality here we can put f of y the distance between f of y and f of y star then here we put the distance between y star and the first order differential at y star and finally the distance between the two first order differential now everything is Lipchitz there so we use the fact that f was a Lipchitz function so here we can put Lipchitz of f y minus y star so here we know that in y star it's actually our good set so in y star our functions the two values were coincided with the common Lipchitz function so this one is actually a differentiable function there and I get a little o of y star minus y actually because this one is the same as the distance between so twice the distance between g of y star minus the linear part of g at y star which now is the usual differential for one function so this one is by the second condition that g was differential of the distance and here we use that again this one is a Lipchitz function it's very simple to see that the Lipchitz constant of the linear part is the same of f so here we have again the Lipchitz constant of f times the distance and we would like to see that all of these so that's our claim is an infinitesimum of the distance between y and x naught so that's what we would like to prove so that's the claim OK, let's see first first this one here so this point y star is a point inside our ball b to r and r was chosen exactly to be the distance between y and x so I have here an extra factor 2 but let's term there first this 2 this term here is at most 2 times r which is exactly y minus x naught so this one is fine now the only thing we need to care the Lipchitz constant is finite of f we need to care what is the distance between y and y star which is the only thing we do not control so this distance here we would like to see that it is infinitesimum with the radius of the ball so why this? here we use the first condition for our point and we use that actually this y could have been outside our set but the space I have outside omega tilde is very small because it was a point of density 1 so here I take a ball around y with radius y minus y star so this is b y minus y star center at y and you see that this one was the point realizing the distance with omega tilde so this one is contained in omega minus omega tilde intersected with the ball b to r so in some sense I have an indirect measure for this distance because I know that the volume of this set is very small so it is contained in the complement of this set here so how can I estimate the distance there? so this one up to a dimensional constant which depends just on the space dimension is exactly like the measure of this ball here to the 1 over m I renormalize for the ball b r so I put here r and here the ratio between this measure the measure of this ball and b to r and here I am changing this constant there is a factor to an extra dimensional factor there but I use now this information here that actually this ball was contained in the dimensional set so that volume all to the power 1 over m so this volume here is less or equal than a constant times r but remember what was r r is the distance between y and x0 so that's r and then I have this ratio here which is b r b to r so we have connected omega minus omega tilde divided by b to r to the power 1 over m and now we are done so omega tilde at x0 density 1 so this means that the complement of omega tilde has density 0 at x0 so this ratio here this ratio here is going to 0 as r goes to 0 that's by property a so all together this one is y minus x0 times in infinitesimal so little o of y minus x0 which is what we wanted to prove because now this one is already a little o of the distance this one as well and this claim is proven that we are summarizing this proof so what we have done was looking first at the points where the function were splitted and then we know that we can apply our classical or edemacher theorem finds two differentiable functions which approximate our function there and then we looked at the point with the multiplicity 2 and we select kind of good points which were points of density 1 and points of differentiability which is the definition living on that set which is what we call g the central point now for that point there with this computation we verify that we have a differential and the differential is exactly twice the linear part of g which I mean turns out to be good with our definition because for a double point we need a differential that's the distance between f and twice this linear function is actually infinitesimal with the point so I conclude the proof here but just few words how to proceed for further clue that's just by induction again so now look at three values we may split in one values 2, 1 and 3 and the only thing you really have to carry is the multiplicity 3 because the 2, 1 you can treat already and the 1 was the classical arithmetic theory and then you proceed up to your q so this one is just part of the proof but all the ideas are somehow here in general the omega 2 is the closed side and in many cases it has zero value yes but in that case I mean you don't care you just measure zero you don't need to prove anything about that set because the final claim of the proposition is that you are differentiable almost everywhere exactly exactly so in that case the classical Rademacher theorem would suffice ok so this one is is kind of an important theorem because we had a notion of differentiability but we need to know that the notions are differentiable because otherwise the notion was kind of empty and now what we know is that a ellipsid function which is nothing else than a soublev function w1 infinity is differentiable almost everywhere and now this one is not difficult to see in the notes I gave the details but as for classical soublev function the differentiability almost everywhere when the exponent of the of the soublev class is not infinity so this one is now a corollary of this of this theorem but let's state as a proposition so every u in the soublev class w1p omega aq is and now here I have to have a little word approximate differentiable so what does it mean let me just explain in words what does it mean approximate so it's not the point wise notion of differentiability like for ellipsid function we cannot claim for soublev functions but for every point you have a set of full measure at that point where your function turns out to be differentiable so it's really kind of the same notion approximate that's a way to see this in terms of measure theory but what is important is somehow that this one allows us to define the differential almost everywhere in the notes you find the proof what is behind this proposition is a losing type approximation theorem you know that the soublev function coincide with ellipsid function on big sets so where it coincide with ellipsid function we have a differential since the set is not the entire set this one will turn out to be just an approximate differential but the proof is actually just a kind of losing type approximation theorem but you find details in the notes so I don't want to spend much time on this the only thing I would like to fix now is so now we know that given such a soublev function we have a differential so let's fix a bit more notation on this so always I called this linear map Li but actually the second condition which says that so this is just a remark on the notation so since Li was equal to Lj whenever fi of x was equal to fj of x so now we make no confusion in writing Li so we can write Li equals to dfi at x in this case our differential in terms of notation takes the usual form which is the first order Taylor expansion of f at y is the superposition of dfi of x0 y minus x0 plus fi of x0 so but let me remark that this one is just a notation so I'm not claiming that each fi is differentiable I say that I call this linear map dfi and I make no confusion because when two fs coincide also the linear map coincide so there the fi would be the same so this one is just a way to take advantage from the notation the map and introduce has to the next point which is how to rewrite the energy at the beginning we introduce this energy and now what I claim is that proposition again is that for w1p omega aq d of u let's say square is exactly the sum of the q values of these linear maps almost everywhere so at the beginning we gave a metric definition of d squared taking this supremum all the composition taking the partial derivative and summing these values now we have a point wise notion of differentiability and actually the claim of the proposition that these two energies coincide so you can take the sum of this norm squared and this one is the Schmitt norm of these matrices squared and we will reconstruct our energy at the beginning in some sense this proposition now I don't have time to give details of the proof but in the notes you find it it's important because it justifies our energy so because when our u was actually the superposition of smooth functions you know that so u1, u2, u3 you know that the right first order of the area functional is this so now what I'm saying is that at least the energy I introduced at the beginning is giving me the right first order expansion in the smooth case and that's a way to see that this quantity is one way to justify the introduction of this quantity here it's exactly the first non-constant terms in the area function and this one close somehow the first part of the theory so let me just summarize what we have seen up to now so we started from this example which was so our leading example we will always draw this picture on the blackboard so the picture of complex branch points would like somehow to write this branch point as a graph of a multivalent function because there is no way to write as a graph of one function but for this multivalent function we need to develop a first order differential calculus and that's what we did so we have a notion of sublet functions we have a notion of point wise differential we have a notion of of modules of this differential in the sense we can start doing the analysis on these objects and try to find solution to some minimization problem or to some PDEs and so that's of course a no smooth analysis because our space is no smooth it's just a metric space with this combinatorical structure but this one is kind of enough to allow to do all the procedure for the for the usual calculus and now what we will start today and probably we will finish tomorrow is to see that actually we can find solution to minimization problem in this context now of course we should look at minimization of the area in principle but that one is a nonlinear problem which is technically more difficult than looking at the linearized equation which is dila plus equation so let's focus on dila plus equation in the proof of the nonlinear case will come in so what we look now is a generalized harmonic functions in this case of of multiple bands function so what we try to do is to define harmonic functions as the minimizers of the Dirichlet energy so let's introduce the generalized Dirichlet energy I mean now we have everything ready will be just the integral of our domain of the u square and let's a definition for this energy we can define the boundary value problem and say that u is harmonic or in this context is called dyr minimizing if this is a solution of this variational problem so the form of definition is u is called dyr minimizing and this one keep in mind is a synonym in this no smooth case of harmonic if the integral of the u square in omega is equal to the minimum of this integral of df squared in omega where the minimum is taken among all the function w12 from omega to aq such that the trace is the same as the one of u so which means so we define the trace in terms of the composition with a distant function so for every t the trace of the composition u t is equal to to the trace of the composition of f so this equality is meant in the usual sense of traces and we test for every with a distance to every point on on the space of q-points and sometimes I don't need I don't know if we need later but we will call we will just write this condition as f on the boundary equal u on the boundary but this one will mean that all the compositions are the same on the boundary so a function minimizing such energy with a given trace is what we call the minimizing and now what are the basic theorems about this minimization problem so at least we need to at least one of them which is the existence of solution so distance of diminimizing we would like somehow to show that we are able to solve this Dirichlet problem for such an edge so given g in w12 omega aq that exist u in the same space w12 aq did minimizing u at the boundary equal g at the boundary of omega t which I mean I remember the boundary we just look at the composition is nothing esotic there so the first theorem is on the existence and then we have a first basic regularity theorem which is how regular are the harmonic function in this harmonic function in the classical setting analytic here they are they are not analytic but they are just just continuous so this one will be first result and the second one is the regularity every u is older continuous with exponent alpha which depends on the dimension and the number of values in the interior so this one is the starting part of the theory so actually we can find solution to this problem and they are somehow more regular than just being sober function they are continuous but nothing more than continuous and the reason why they are nothing more than continuous you see also with the examples of harmonic of complex varieties because the complex varieties they are examples of minimizing currents but actually there are also examples of de minimizing function where they can be written as a multiple covering of a plane so this one of course is more than older continuous but we we could have parameterized this complex variety with respect to the wrong plane in that case what we see it's something like this which is continuous but here it's not differentiable because the slope here is going to infinity this one is kind of the same curve parameterized with respect to the wrong plane this one gives an example of a de minimizing function and it is just older continuous so also this result here is sharp no so this one for example is an example where it is not liptured it looks like so w here it's like equal plus or minus I mean you have a different number of comments but it looks like z to the two thirds here you have three sheets coming there now we have just few minutes left so let me just comment on this first part and then tomorrow we will start with the regularity part and the first part I mean now I can go somehow very fast because it's an application of the direct method in the calculus of variation which Alessio spoke yesterday so for this class of function we have all the ingredients and for this energy we have all the ingredients for applying this method so these are really the last comments so more than a proof will be just comments on the existence of de minimizing and the point is that one can apply in some sense the existence the use via the the direct in the calculus of variations and in particular the ingredients where that we have the compactness the sequentially weak compactness for sequences of function which means whenever we have fk sequence of function with the trace of fk at the boundary fixed and the energy equibounded then there exist a subsequence and an F such that we have convergence in L2 so this one is the classical compact sobolev abandoned so sobolev norm I can find the subsequence converging L2 the proof is very elementary and just pass through the composition because this sobolev function are defined in terms of composition with the distance so for every composition I can apply kind of the same argument and then pass to the limit so it's not 100% exactly like this but the basic idea is this the second element is that actually the trace is continuous in this limiting process so let me write here and actually the trace is kept and that's again the trace operator for classical sobolev function was continuous under weak convergence and here I mean the trace is defined already in terms of composition with the Lipschitz Bapp so this one pass in a natural way into the limit and the second element was the lower semi-continuity under this this sequentially weak convergence lower semi-continuity of the Dirichlet energy so which means that the Dirichlet energy of the limit is the Dirichlet energy of the sequences as cables to be very few comments also about in this point in the notes you find that the proof also of these also here the ingredients are very elementary and to prove this lower semi-continuity one of the best way is to use the first definition I gave of df squared which was the supremum of all the norms of this partial derivative on the composition of functions then of course this one is a supremum on a countable set but by the approximation you can reduce as a limit on a supremum on a finite set and now for a finite set you have on each set composition but on that set that composition is converging weekly so you may apply the usual lower semi-continuity and then some back to reconstruct the entire energy so again also at this point the philosophy is that since the energy is defined in terms of composition and the compositions are lower semi-continues then this property remains as a function of course you have to pay a bit attention because the lower semi-continuity you have to test on open set so you have to approximate properly your partition where you put the different compositions but apart from these technical details here the proof is really the classical one so df is the soup of a composition of leptin functions so the graded of regular sovereign functions so once you have this ingredient with the mechanism as I explained yesterday so you take a minimized sequence you have compartments so you extract a converging subsequence and the energy of this subsequence will be the minimum energy so you solve the existence of this the minimizing function for today I will stop here and I will give you time for questions and tomorrow we will start seeing some details on this regularity result that is functional a bit more than so but they are continuous and that's actually the best we can do so if you have questions that's a so that's I don't know they should be on the web page so they are already there other questions so let's have the coffee break