2/3 The rise of sc-retracts


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2/3 The rise of sc-retracts
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Fish, Joel
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Institut des Hautes Études Scientifiques (IHÉS)
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In this talk, we discuss the second of two fundamental analysis concepts polyfold theory is built on: sc-retracts. In particular, we discuss how they arise naturally as a means of using pre-gluing maps to parametrize a neighborhood of nodal and non-nodal (or broken and unbroken) maps near a nodal (or broken) map. Despite locally varying dimensions, such retracts support a version of the sc-calculus on which the chain rule holds, and we define M-polyfolds (manifold-like polyfolds) to be those topological spaces locally modeled on such retracts. [Related literature: Sections 2.3 and 5.1 of Polyfolds: A First and Second Look.]
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that thank you are so so 1st right off the bat I want apologize for this juggling of the schedule but as a personal obligations would just make it difficult to schedule so hopefully this will be the last of these US switches but knows this because I'm OK but right I want so people pointed out a couple of things to me after my talk yesterday that I wanted to bring up there was sort of funny because of these peoples raised exactly the sort of pointed I've raised with a helmet on a number of occasions which is it's this oftentimes you start talking about this stuff and there's just ambiguity in the language and but once you sort of brainwashed enough to sort of understand what's going on in that ambiguity become sort of natural that I committed those same since yesterday off and so I kind of wanted to to point out something about that so there's this ambiguity at it in terms of how I write it into the editors acknowledged to be the right to some extent in terms of the which is meant to be a scale 1 space ending the nite which is just a level In particular the base level 0 level of that as space I mean in particular are you you this looks suddenly confusing and but what is this really mean by as Kazuhiko is scale but states that is the whole sequence of scope of Lenox spaces following by the status of an open sat right on are you sitting inside it and so do the ambiguity is sort of always meant that he is both men what I write he it's meant to to mean both scale that the whole set about space is very time easier ascent statement like this I'm talking about the 0 level by itself open sets in particular in the scale by space always means you take an open sat on the base level and it has the scale structure induced by taking in the intersection of an open set with all the other 1 spaces and all the other of 1 spaces in the sky 1 expects to make sense so hopefully hopefully at that clears things up and it's just I mean he's a priori more you go with sort of stuff you the more you discover that you just need sort of more and more notation and then at some point it just gets too difficult to keep track of all the notation you need to allow persona ambiguity to sort of actually make anything understandable so on that seems to be my preference so it place against stock even with the sort of basic questions could believe me the 1st 10 times over the stuff this is exactly the sort of thing would drive me nuts please ask but I just can't handle this the 2 2 this is something you use the same thing it should also be living there will be a case of its equivalent enough I need to see a precise definition questions OK so the 1st thing I really wanna do during today's topic is just recap a yesterday Israel's rebelled on sold the 1st I would say it's remain thing that I wanted to point out was that the action of the regulation group is not classically smooth so had this sort of toy probably show that showed up in transition Matamoros setting up sort of try and set about manifolds remorse analogy and then I sort of thought and I said the same sort of action shows up Ingram of witnesses essentially model it probably won't consider action repressurization is not classed as smooth was pointed out on the on the total space in the ambient space of functions we wanna work with is a smooth action if your body for Ardelis spaces cut out transversely of any and action is just restricted to that space so that was a OK will want that we would really like to have his action on his total ambient space to be slowed in some sense so we introduced scale bonnet spaces in scale .period different ability as a consequence of that we had served 2 key facts what is the Nour the reprivatization action is is the smooth and is also the case the chain rule holds general holding basically means that we now actually have this new notion of differential calculus substance right arm and then repressurization acting smoothly is nice because that meant you could build transition charts some notion of smoothness between them so you could build something like the scale about manifold and toy case cases and last thing I want to point I did mention last time but I'm you know for me if someone tells me the function is smooth that always means the Infiniti but I know there's some collection of people who also think that moves should just mean all rather smooth function should be see infinity art of fermion above there's a number of people who think of smooth function to just be C 1 and so for all of my talks move is meant to mean infinity right in particular as the smooth is meant to be a the Infiniti not as 1 defense and they concern say Well you actually show the average transition actions he slowed the answer there should be speaking no but this proved carries over into Serbian radio and then and you can actually prove that the reprivatization actually isn't quite the Twinkies that presented is in fact the questions about that some new material today we want to French fries and should stick with it's status we want charts here notable would say maps and we're thinking in some sort of a gram of winning when setting afterward that cost me about borders in groups and we just 1 sort of understand the charts near-normal maps should be so the image of your map here were thinking of looking something like that 2 spheres maps from from amounts from other nodal sphere audiences are manifold Aurora departure Wednesday for instance and so even the sort of state the status were kind of replied implicitly assuming something in the background here that we we know it makes sense to me to be near a no map in a reasonable sense so where we already were all I'm already sort of assuming that we have some sense of what it means to to put a topology spaces of nodal maps in this largest base in particular In particular would like to be the case that this no Nomad is safe for instance this man which is not known you glued a little bit like to be the case that this sort of close to this that selenium a near and that in particular is 41 trying find a chart for charts a for instance is centered at the sort of man Seoul so that's main goal for 2 days of and how this appears particularly in and out of the political framework In end so the 1st step guess to take this picture here and for simplicity we really wanna cut away as much of the topology of the problem is as as we can and if we do that then that's returns it into some problem that looks when drawing on the flight is a little bit poor so my apologies for that I hope Joe's delirious but I hope you know but I hope help the lease this is the fact that can period so this is what so this is what I wanted to I 1 toy chopper wasted apology on and turn it into this sort of problem here were received with pictures on and then more precise statements on and on how this is going to be and being useful for us so but I would like to do know is that it is a little warmer problem around the world and give warm-up which goes like this definition for fixed Delta bigger than 0 and a natural number
define each paying Delta are ladies bribe subscripts rather to denote coordinates on these guys just as it's convenient to do this if I want to be precise hundreds of so I H K equal to W 2 I am and I wanted to find this to be equal to the set of all as in take a bloke such that the of F times To the Delta absolute value as used in L 2 4 far between 0 and Cathay and then thank you under In the norm In this case is the sum of culture in absolute value between 0 and K 2 in a row or costs 1 each of the 2 girls S In the Gulf have squared the ideas and what I wanna do is stay in all work for 0 strictly Western Delta 0 scrupulously don't want stability and ought to do and I'm going to say all this less than 2 high the show age can they will still rise the switchover Jesus obtained defined to be equal to inch suitcase Delta Chi is off city about so there's some hints in the lecture notes the have alignment and of course you can ask made his work through this and I think yet sold so that's an option as well on that is the key steps so that's the key step and I guess with the stomach hint hard-hit by providing the notes is that this is the sort of exponential decay in particular the fact the have increasing waits in terms of exponential decay is crucial to To guaranteeing that compactness fizzled however I would add a sort of an addendum I would as an open-ended question I would just sort of say you know explore what other possibilities yeah well possibilities 1 might have in particular you know it is for instance important the delta's no God to infinity but do you need exponential decay sorry the Delta's increase when strictly increased from government and all that would also work so that raises also natural question what I had to apply here this you sort of completely strange and you can choose a different Icap there if you like because I'm talking about problems Ingram of Whitten this to and that showing up as being relevant and then for those who done sort of more analysis in the subject of this basically this number here has to do with the spectral gap on operate appropriate support operator into prices shows up in a grove of which case and if working inserted as of tearful acknowledges something these wages these caps sort of have to be are different and so there will be sought a corresponding changes but in any case this is sort of a predecessor the prototypical scale 1 space that occurs here a lot of Polly fold down a lot a lot of powerful literature I think and everything is sort of modifications on this thing good OK so now what are we gonna do this interesting of lies what else could it be that it is the 1st starting from what starting from where Starting from here I would say this is the obvious OK so why is this the choice so I would say that the point is that you have a non compact domain if you want have scaled 1 of space they need contact of higher levels into lower levels and you can't just you need some sort of you need some sort of increasing the regularity alone won't do it needs some additional information to the exponential weights await the exponential ways that we put on certain heavily weighed things on the outside that in that value you can tinker around with it that you can tinker around with it and sort of see that guarantees Compaq embeddings another way to think of it is that in some sense morally those exponential it's kind of force in many ways allows treat the problem is if you're dealing with concern about an infinite domains for about abounded and we would not know that you would wanna cut off and has not yet so again so that that my my my addendum to my homework problems explore the necessity of this sort of exponential right in Seoul began in so the solution is basically doesn't have to be exponential could be a lot weaker it could be a lot stronger but there is that there is an important reason important key in there but you choose exponential decay I think 1 of the reasons you choose exponential is that this place very nicely with the corresponding asymptotic analysis this is 1st time this is the result of an imminent of mice for that sort of thing you have to listen to the views of of the water in the history of the just wanna constructive space you have a lot of freedom but if constructive space that's useful for applications that you will be more careful in the alternate follows form with Robert question the reason for the community and the bomb for this definition the most natural thing to think of as is swallowing drop remains here in it right now and then and then hopefully for and sold OK all drawn solely Drolet I erased so just have to buy time on and that as better judgment I expect from it is supposed to have cylinder so here is that we have right side said that the picture areas previously we had a noble Kirby my syndrome of Whitman said I said OK I would forget about the parts the topology of much possible reduces sort of this Paris noble despair basic right so this is what I have is that we do this it will look this is a despot I had this noble .period I want treated like a puncture and solitary like a puncture whole for coordinates which triggered a sort of a positive hats of the citizens of being part lost costs 1 off and on over here this is going to be using the winners from the opposite direction or minus process as and now any like if this is your domain names now this is your domain you have a map to find honestly this report that Battista maps on either of these 2 all 2 cylinders right In a particularly have sought an asymptotic matching condition in this so then we want to do right because our goal for the day although erased it up our goal is to essentially trying to find parts for neighborhoods of nodal Kurds we have to do what will be the see where it's in my notes and the right thing to do is to
is that that's going to involve a sort of twiddling maps and so I have to define what those aren't for you and so soil at the start of the picture and so so here's this picture so the idea is the idea right if you have sort of it had the note the noble map the nodal does matter and then you pretty low you know you know you end up with models and so that's the picture the drawing here and in fact and in name it instantly called outstanding names EA and I am going to define its very precisely on the border as the area is equal to this 0 ,comma along with an escort of processed ones this joint union lines or the 0 this hasn't as prime quarter says he cornered us 1 is hesitant prime Coronet pointing that destroying unions and I want people to know this equivalence relationship which identifies these points St With this prime plus lost all are common to the prime plus Vader where are people's needs to be won over absolute value of a minus the and in days equals mile a Eatonton minus to hide by the so what am I doing here right so I know that this is significantly different than what's done in the McDuff sound about right at the idea being that you know if you haven't noticed a hole of recurring modifying the at occurs for this sort of argument that you make the argument says you you find this is previewing map which allows you sort of construct nearby Mount spider bite from the little ones of provided you've given me this complex going parameter which I'm calling area which is also the previewing that hasn't been written down I'm about to do that was so so I'm using catches from analogy this is this is worse required this from the idea being my understanding of captains ideas you cannot be a firearm is Is that Is it in something like our Gromov wedding or or you know in the various arms mild Marja was not there's a glowing map and the glowing that should be understood as you take a broken solution to your problem and the glowing that takes you from that broken solution possibly forever to another solution Free going says give it to Our priori non solutions but put them in a function space that's close to where the solutions should lie so you be it's called Regling because usually outlined standing is that you take solutions to this noble problem broken problem you Kriegel those solutions together and then you run sort of a card iteration Alabama fixed-point arguments to basically say there has to be in the nearby solution it is that roughly correct Kateryna read dozens of this was the last 2 bullets were alerted to watch the news so we have a good reference I think is located on the deserted standard language to understand our focus period OK so right and so what's going on here so we have to define at Ringling that but in order to define that revealing that needed domain that map in order to define domain that map well you have to do a bit of this this this this type of glowing here on your domains and then we're going to find maps on these on on on this new domain so do that now so people OK so I was on suddenly precisely given a and C and really thinking of blood a being near 0 then give you plus or minus as maps from Paul plus-minus across as this 1 PM to 4 convenience again just working are too when we can change things later if necessary although the accident is interesting tricks and we have previewing maps of the plus minus well a it's defined as a map from Zia into to end so you can grammar and then to abuse to these maps here and I construct a map From the start of finite cylinder into are and and it's going to be given by pinup West saying the plus minus the 2 beta test when it's all over till you twice as the team played 1 minus the players 2 at times you might as minors are you in the where the wounded in where data essentially has fallen form it's a kind of function this theater of S here so that's has director has compact support 2 the prime is less 3 0 3 the In the best plus a bit of mine asking for 1 majestically so what is being defined this going mad for previewing math rather is well-defined this of course if you haven't done any Quigley analysis of afforded something like this looks really unpleasant but if you have been doing analysis of you can listen and see what's going on here I think hopefully fairly clearly basically what you're doing undecided got up you have a cut of function and over here you have 1 minus a cut of function and so what that means abandoning done this you can tinker with it if you haven't done before but is the essentially interpolating between you plus you might all module the sort of ships in the corresponding
domains and then the picture maybe the picture that's essentially understands what happened catch nitrogen this picture of a similar picture of ordering her lecture the idea basically did you take 2 domains said the you you shift them when you add in this relative twist on and then and then you 1 0 identified service truncated region which would be defined to be the and in the CAC on the far left hand side of you Plus which is defined in the top part and that on the right-hand side of Zia it's it's going to be you minors which is defined on the on the bottom that makes sense I think you have a is just that this is so this is a yes this is just up 3 going on and everything done here is is has to smooth for anywhere from 0 new known mechanism I have to think about what's new this means in this case so is a lot of strange least because a new domains of your maps a sort of changing to build a space where it makes sense to make it 2 2 even compare themselves but so will make it clear that some of the students good and so on we well no I mean how do you compare so suppose I take supposed attempt on my life fix suppose even fix you plus a new minus on and then I I compare like other equals 1 and pulled 1 pass by the point is that if you look at the definition of z like your domain is changing right in fact the way I would do it carefully to find on the way carefully defines the even if we consider like a equals applying compared to opt out of the right on PTI over to compared to the to the and say I say for instance right even in this case with this careful definition you're domains changed right all the dimorphic Accorsi ministers Tennessee models with domain of changes so all I meant to say was was administers a smooth at this point already because these things there you need to delay the function the changing of compared to functions you want them to have the same domain any questions smell it's no 1 saying I'm trying to keep this the people the world is looking at the bottom of the role of this move so is mean so as as a ghost and so would a equals 0 it's as if you're your maps have been glued at all so is the noble man and so then what a controls is why controls in some sense how you construct it essentially controls how you construct is the which tells you how to construct the modulus of his neck in between and any more than that of the models of the neck at this at this notice was waiting for a while because of because well for 1 reason if you if you if you didn't twist are they have boundary when Mumford space services would have boundary you know you've done something wrong right I mean there's just I think if you wanna get all nearby maps you have to you have to include with the half of that was for OK so here's some ideas so the the 1st ideas and the right idea but bonds yeah I guess I should say I should be I should say as a captain's told us about a million times and it was only yesterday that sort of the light bulb went off in my head but this was really the right really the right right idea off and that his sauteed takes a while for stuff to get in the history of the idea was to use his free amount to the final a neighborhood of I thought they nodal maps write and added I bring this up specifically because during users talk you know summonses have always signal was this topology on you know on on the space of suitable work experience something right 120 compact by was this topology and immediate answer was sort of like all that stuff I don't talk about that and and that's it that's a perfectly valid reasons not because it you know you write down because the way he would define the topologies essentially viewed role compactness and redevelop the definition of Brokeback is anywhere and it's complicated and and insult you kind of don't wanna do that however if you look at the pre going mad if you look at the prickly map on the on the ambient space the trying to build that it essentially when you restricted to your my July space on it gives you the gram topology and it's not so difficult to see How the Springland that ought to give you sort of neighboring curves right asserted sale can take a noble map and I gave a glowing parameter and I find this sort of approximate non-metal curves nearby and that gives you buy a bunch of sets in that bunch of sets defines a apology for you as long as everything from opening suitable so you know it's sort of a suitable sets and setting a very clean weighted sort of defined the topology or what you expected topology be to be instrumental in space so this is a good idea so here sort of have to be a little careful it's it's it's right but wrong on this following idea which is to you is not this to build a charts for these neighborhoods and so I sit on it just like it was yesterday and make them more precise but I mean with the 2nd statement this will be useful light of the well I mean well but saying via I have a map from sea across the In 2 capitals the E and so on see takes as input Bay which is a going forever and wrong use you know see reality were thinking suggesting an open-air would 0 but just for convenience limits AC on and then we and consists of these pairs you plus and minus and then we wanted to take these 2 the image by the prevailing and that I would define its functions spaces for you so equals the sum of all areas plus-minus defined on plus-minus crisis 1 sorry capitals is another function space which having find isn't busy defining the first one such that there exists a constancy In to end such that the trade clearly tilted not absolute value as d telephone you plus spoilers -minus C in corresponding held 4 well for absolute value 0 . 3 and is the union of a man and see Oregon being slaughtered
penetration H 3 maps from the CIA to onto
it what are you going 201 regardless
of all right what kind of logic is it's it's it's this by definition I mean hot in this and I mean so so this is about these all are each 1 of these is set to continue union over South Korea another set itself it's a sad at the moment right but you can say it but when you can sort of see that what what should what should Xabier any initial context needed actually capped off a domains by these words of did well then we would think these being the sort of function space from a bunch of different a bunch of different I'm Kriegel odds yeah a bunch of 3 glued ramen services basically right a particularly France additional mark .period to modulus can change right and so and so now you just you haven't you had this very large function space of you know H 3 maps from each 1 of these pre glued on romance services but they each have a different glowing parameters for each thought is completely different that makes sense 1st of all the Member contained in each other soldier I mean I don't think it matters that much but Marion running British 20 of flight of the day right that's true yeah that's true no further questions here and there is taking 1 of the founders of the city all right because I want to try and follow the growth of wood and paper as closely as possible and I they do that because we these these functions here then are end up being an age 3 and then that guarantees of the C 1 and C what is important was you 1 put .period transverse constraints the sort of transit service further questions and the question of how much greater because the or refer larger Watson's hot days ahead 5 3 4 per hour as it is number 1 yeah I delivered earlier this year the worst of all I would imagine that that might change things in terms of this here there's a lot of the media yeah I mean I wouldn't be surprised tho it somewhere along the lines making a choice forces you to do some extra work is a safe for instance proving convergence in terms of growth compactness now you have to make sure that things converging sort of you know that you know it's not enough to even sort of convergence herbal regularity that merger hiring Larry I mean I don't know how the argument goes exactly the powerful framework at you know it is not clear is not clear to me I think it but in general in general but the point is you want this to be as low regularity as possible just to make sure your capturing only occurred because otherwise you run into the same mistake are suggested which is put a double exponential in our priori assured excluding something so keep things as a kid you know by making this the larger restricting your ambient space in the interest in general being cavalier about this he might lose information in this particular case idea right probably it's not a problem the original stories you or you could be with me on this which just like being at a display of what would you feel have to do this sort of thing and experience in order to use a but there's also also what happened was up front so we start with a paranormal desks and then hour from now you have brought up 4 quarters each 1 of them the effects and that gives rise to these odd to have cylinders and then those 2 have cylinders possibly forever a gives users EA which is the services cylinder of finite modulus arm and then from that we once we had this sort of domain we also want to find if we had maps to find on a pair of noble disks we wanna know what's the cost but which the corresponding be defined as the area and we ought we arrived at that with this formula right here right and then now cost now of course we said there's some there's some ideas running around here and so what is to use this really that define neighborhood of a noble map and so what I'm saying is you know in some sense this this was a sort of part of our neighborhood of this part dozens big party neighborhood about of a of a noble maps on them and I worry Bielsa to make this this precise here right now and we had a 2nd idea which is a right but wrong which for the most part we should think of is wrong and I'll show you why I is to use this corresponding recalling that to sort of say that I mean if the sprinkling map defies a topology that essentially telling you that you finding all nearby you find all nearby you know not noble curves to a given Noble 1 using the going to we noted is divided apology for us right which it does the new finding all nearby map so why not try to make use of his of his pre glowing than actually just parameterize all your name and your nearby problems or sought all your nearby nominal maps and all the things that now correct correct so what which of my statements was and is now confusing the lack of clarity I think there is a need to know how you do it and distill the ambiance that is not the case right yeah I mean you have to to get in there and you have to take status of the you have added this month and that's true although I would say the most of today's talk sister triumph again as much topology is possible in sort of a bit of a neighborhood the name of a neighborhood that there will lack of love from what the writer Munich could change my definition from a pair of to 2 disks this is the obvious that I wanted to say was that many of those kind of conditions of life in
different areas of the different ones no I mean you can see from me you can see from the arm of you can see from this this this definition of sort of free glowing here is this is purely a local it's purely local phenomenon in you know they're always sort of separated and you know in a sort of on you know they're always sort of been about a distance away from each other as traffic in tactical opposition tells you many people more ,comma peaceful that's a different question so at this point all work beyond decisions this issue right the nothing's home-office and building ambiance spaces right and in some sense that's all we're doing this entire 1st week as building ambient space right in with the appropriate bundles instructions for and so benefit is the yeah we don't run into the sort of you know I you know we don't run into these problems of what associative it in terms of going for instance good this is name way to have this year is only half alright so what's going on here so that that's actually good .period so if you look at this or if I don't put this sort of see here than this function space is essentially the same 1 is the practice 1 mean usual truncating your domain to surpass cylinders so why is it the case that we have is this here and what is that do forest well what that does for us is that if you if you think about if you agree if you think OK I've got nodal map Osoria Noble the parable disks as my domain and I think about maps from there for simplicity was writing down problems you deserve assumes a figure you're basing a problem when when the art that the image of that notice on the basis of their base map is going to go to 0 but then you're gonna want that node to move around and that's what this C allows for basic rights and so the estimate of our try construct purely toy problems for this lecture I could have said OK outlets killed spy really wanted things as close as possible to grow up with because that's sort of a think I think that should enable me to you in a recital of stuff that I've done and even write down charts in that case so so now I will address this .period here about which is the 2nd 1 is right but wrong and I want emphasize the moment this idea is wrong for sure and so why is this idea wrong for sure In history that's exactly right Saskatchewan mentioned as art yesterday as well but if you when you look at this problem here I'm all when you when you try to do this well if you want if you want to use the previous Prego enough to build a charter or privatization what you wanted at least to be you know you've at the very least you'd want to be injected and and we have a problem that because of the way weight the scribbling that defined sort of truncates know truncates domains or lose a ton of information on and and so consequently this map and of being infinity to 1 In general general a non-zero right and so Infinity Wu infinity infinity to 1 that was a terrible idea to use as a proud as a prime tries in the final days the industry is still using the removes the the absolutely worst mine "quotation mark considered to be of pirate went on so when you move down so that we can still see this this definition up here you can see the wants what's going on here right you know hello that's been here is new plaster you pluses defined say certainly on this region hearing and this data in this region and this region he originally said is 1 so he was given 1 but minus All-Star vote but what might beta is 0 in this region soda map to find in this region on this redirect here is just you what the same arguments are tells it on this region here it's just you minus but with just you minus then you plus is defined you defined all of this region here so all this information beyond just being killed is gone being killed because this is a cut off function it but you know it looks like this but it cuts off you plus after some finite amount of time and we just whatever whatever the map is in this region right the offense good book In the end of the season there's no way to to stop the conventional From this from this sample done so far in Europe but you're absolutely forced to lose information you always you always have to truncate here mean how how can you know if he's always communicate use information from the SEC right now I have a good history of this this is another way of making the announcement which you don't use them this is the reason scale and greatly by the whole entire day on news that it is about to happen to them but this has that to do so puzzle was probably that of the question all problems In fact he perspective and well that's something so I said I so I said right that wrong and now we see Riau is this idea here is wrong so then the question here is why is a kind of right so so the key that the key to keeping it fails here is that you in this set up anyway you lose a bunch of information and Southerners of text that 1 would like to try to do and that idea it is too find a way to keep track of lost information and so the wayward do that enumerators at to prepare so I'm going to fire in August we find defined another cylinder right this is a
W in cylinder and tried to drive from suitable parallel to the other ones because it's related chemical Coliseum actually going we fix that if I were only considering this finite portion of the cylinder this is essentially going to be easier but I want consider sort of the whole thing the whole thing is called C associated so I find under is CA is my doubly infamous owner I can write a definition for CA and you can even serve you should be able to guess it from the Foley seen so far or lost however this it's going to be destroyed Union for as prime moment cross 1 the prime model of groans relationship which is the same as the 1 we had for this team is related to prime plus our kind of faded and then and I can define faded players going sorry my schooling or other miners previewing what is it into pre-clearance were growing has the following sure the of panel where the money almost all of the issues that will the you could named 1 of the that support I'm not around me that that was what he did and I have to write that we don't have all the world right and what I'm trying to do is is motivate under what what here's what's OK so the goal of my talk which interrupted from time here is is to sort of say Look here is sort of a sequence of things and I want introduce the prevailing map which is sort of standard in this sort of analysis right so United United away from doing some sort of revealing that OK and know I would do is say OK well How will we like to do is to use them out to make a privatization sort of nearby something noble and it there then the reason you can't do that is there's information loss or do keep track of that information and then the punch line of this is that if I keep track of that information in kind of a clever way in the clever ways sort of the way they used to be easy to find it what you can do is in this in this way that happening is in his weirds subset of a scale 1 of space as it turns out it's it's an incredibly strange subset it's very difficult I mean just adding it anyway it's fairly difficult jumping dimensions and so forth but nevertheless it hasn't straightforward smooth structure on it and this provides this then with this you can then build essentially something like a manifold right which is a smooth manifold as the smooth manifold in some sense which has all but which has enough structure 2 1 has differential structure you can build a friend on periodic Central and sold the U.S. Census yes because because because because that's what you're suggesting is understate what next have you build a friend theory on your problem how do you prove perturbation theory on how you prove regularization would it ever with any other option is to try to do the poisoned following a story so I know all that stuff is going to appear if you want to make a change at some point that is OK Uganda thousand pages to write that some units that's that massive enough however to I would think that due probably has more polite and you have said this year's said while in the last year of life and don't worry about it Universal by saying that none of the Member of the Standing used solely on the grounds of a home in the day story of this country the value of the company conditions can they were in the lead you have to use it in the past and minimize the of look at the hospital had shown some of the more than than half of the people in this section will that would mean a lot to do that in a way would that the right and 1 that will a the where is this going to be here for the views of Ferdinand and most of this minors should factor through of this becomes a plus and Dictionary of applause this is a minus In the my wife and I think it was a so if you want to however modesty and last year's there is always the Elisabetta now you're right you're right you want and I know that I'm fed up with the weather was 1 "quotation mark part of that payment is wasting time and this was a lobbyist I'm getting their rights not so so so yet interview singling out wriggling before they should look terrible and and so I want to do is try and make this a little bit more understandable so this looks awful on you know the sort of nowhere around the fact that it looks awful but you know once you can open it up as you see where it comes from but it's it it makes a decent amount of sense yes the 1st of the year of amendments to the law of the land and the priorities of he said it is likely he made his way through the idea there is enjoyment makes life a lot easier right so Horowitz said alright I guess the 1st claim is a sort of this is that this keeps track of certain lost information on at that precise in his 2nd on if you kind of wonder forget about the fact was the 1 big civil occasional citizen just the 2nd is that if we just seeing this for the 1st time just kill these terms is the average value terms just just pretend they're not there and write the same thing down and at least gives you a toy problem to tinker around with right onto the why those terms in United's 2nd on the active in the middle of right above it it's clearly different now said some farmers and the others there is a lot of symmetry the idea is to keep track of precisely the lost information so resin once ITC buried 1 of the 1 been years and see one-liners buried 1 the biggest so many others symmetry is designed precisely to keep track of what information I can show you how it was that this is moment that had it's giving it it's keeping track of the portions that were killed by you ,comma functions previously by the head of an international the point is that you the point is that you take the point is the interplay between the sort you have it you have it you have a domain which you might think yet to domains splitting the 2 pieces and a prickly Napster keeps track of this information and disinformation interpolate nearby so the Midas schooling that keeps track of this information in this information and an interpolated in between and my claim is that you can actually reconstruct the 1st 2 maps from the latter 2 many suggestions music and promptly at the user level of management in line with the and and of course I'm to be assaulted questions as well as follows the was the voice upstairs sold its interest in the this isn't whole from infants and older model was upstairs trading ,comma tho taller despite occasional
I'm so so here's what so here's a trick and this trick is really I think sort of where this comes from so writers as a matrix would keep track of both pieces simultaneously and if I do that well I have I have they disobeyed one-liners business today understanding of a loan the demand 0 0 tie is the shift correspondent saying new plus minus what 0 1 months to the day it is it's not until you have guessed that area of S & steeple to minors or over to and the shift mask from any applied to some math you value with the best teams is that you love as much as are t-minus theater assuming they have this pluses and minuses the thinking goes in much writing it this way I think is 1 1st episode verified sort of his true after you had sort of equivalence way 2nd this ownership avian here Frank Segars said Sunday the 0 2 terms of 0 efforts were offered twice problem and if you do that just had his matrix and the 1st thing I want to point out is that this thing is just as obviously invertible right because computer determined and it has to be 1 because these things are kind of functions and the weight properties this ,comma functions have so you invert and then this is then this is now operated here about my daddy and I've got the shift operator which is also clearly invertible so right it this way it's clearly invertible I think Freddie fixed at it indicates touch more working again and again and again in a lecture notes on their home problem provided hints which Elida lighter walked through that this is still my ejection even with these terms entitling nearby ejection so so this so I think what this sort of with this sort of does in some sense anyways cleans up this next I lied I prefer to be written this way OMB and then so that I can say on but I can say the source the election of smiling this moral for just a couple I think this year the area right now so I thought I said I thought I said it once I consider again the purpose of the average in terms that is to allow on is to allow right as a question of front why we had this constancy probably function space is just erased a good writer but not here we won't allow these 1 allowed as this this constancy in here because you wanted to it wanted you want it to be the case that you know can move around right and so consequently when you put in when you write down these already right on this matter you need these averaging terms to actually allow you to allow it to be the case that you can hear you still allowed for the case or that note can move around so you'll see why report this is list of the right because essentially your your warranty and expects the so I thought I wanted to be considering so so here's so here's basically what happens and why survey of this relevant for anything so we had this idea so we had this idea that we would like to use this plus going to build a chart for our neighborhoods In the problem we had was that it was 1 so there awards keep track of his lost informations nite keep track of the of the of the plus schooling the minors is going and we said about was objections and that's good because what that means there is a device that all to be equal to the set of all triple century you plus you miners such that of such that all minus a few plus minus 0 then the plot spooling maps essentially 0 into In 2 intensity my objectively this is for any Connecticut this year and for so now these we have objective correspondence and what had wanted to correspond to the next natural questions to say Well is it possible To give this space any sort of differential structure and you know you look at this service OK the zeros of this map will the zeros of of the otherwise cleanup you must still be some sunsets who knows what that looks like right now Opry or it might be complicated and so why should that have any sort of differential structure and the magic is is that it has and in a rather it say it supports please see calculus in just a brief 2nd I sort I can sort of tell you why this matter here we can we can call this map here of but but In the box not doing on you last year time and they're always said was what this thing is invertible there's no way you can do is you can define this matter are of a year plus you might as
equals I am so it's basically you take box not going to compose that will serve the projection to the 1st factor composed with box .period the inverse right and this but the story otherwise this is on tape In was you -minus if a student is 0 and not the 0 to define this mad and it hasn't been the case just to go over this very quickly here is that this map has a very nice property which is or composed R is equal to all and it's SEC moved endit a consequence of this it's very fast on the consequence which I would go over was already 5 minutes of my time on but is that as a consequence of these 2 simple facts you can then define notion of of the nastiest would function defined and unsavory image of all and the image of auras precisely a set of points where their demise previewing is 0 you don't you do that simply by saying well after flying from say all into some other spaces what I wanna say well and any other space all prime is as the smooth precisely for SEK precisely if ever composed with all our on a slightly larger space is SEK and so the boys you this this or acts as a retraction and images of attractions recalled retracts and if there also and BSE smooth than that guarantees that deserves support as calculus talk about is more in my in my is the beginning my lecture tomorrow but the whole point of the strike the whole point of this what is essentially writes a lot yes I can summarise very briefly we said but we said that we will we really want to do is have a parameterization rich hot nearby a noble maps and all we did is a wall the prevailing Napa's infinity the 1 so we you we made use of his my going maps in order to cut out a bunch of garbage and make that not 1 the 1 but now I said is the clever with his rewriting you can see that the bats that can be written as the image of this as his mood retraction which will talk about more next time and as a consequence supports the the calculus meaning we have a notion of a differential maps from 1 set of this form to another which then these then divided the models of local models for Empoli folds but we'll talk more about this next time time we all understand the questions of the people in the end we have been there's no limit on how many questions the fact that of the matter all the move to have a way a lot of this is the 4th of year the 1st time when it the company something more general but on the bottom and for those with true this part of the more than 100 years the two-year-old because you think you know you have to go out there and show that we used to say that this is going to have one-dimensional the much qualified so the have the jury the question again this year was over a projection of the 1st factor so if you have some but you through 0 up the 2nd agenda said that the all 1 of Texas wide he was Texas 0 what's the image of all our or rifle R & B this would well retracted precisely 0 this the is a subset of it's a subset of across the all make more clear next time I went really fast in the last year in the last few minutes and but the claim as their own the claim is that the image of all our is all tinker with it I think 1 thing not all the on way to the end of the of the war always In the process of should be a subset of secrecy the ways of the of American something which agree some of the things that have had to so if you cross the dissolution of the recovery was it's about education problems course the this it was just about the biggest factor in the opposing side see cross years not the colonel it's the domain of a map the domain of bomb that you're still going to be the issue followed almost the followers of civilians the year for big stadia in front of the close a 0 0 tie there was a long series of you know the idea that political views of those who have a history of this will change the quality of the rest of the is 0 growth this year just it's just what might then all sold by a school unless a deal is final year might going when is equal to 0 over the 3 year I have president of the Commission the former CIA of year world but this is 1 of the reasons why the invisibility in the Hollywood that I do actually have to go so far as to


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