3/3 Boundary, corners, strong bundles, and implicit function theorems
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Title 
3/3 Boundary, corners, strong bundles, and implicit function theorems

Title of Series  
Number of Parts 
36

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License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

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Subject Area  
Abstract 
In this talk, we generalize the notion of scretracts to include cases with boundary and corner structure. In addition, we develop the notion of a strong bundle (of which the CauchyRiemann operator is a section) and state an implicit function theorem for transverse Fredholm sections with compact zeroset, which guarantees the zero set of the section is a manifold with boundary and corners, with boundary/corner structure induced from the ambient Mpolyfold. [Related literature: Sections 5.2, 5.3, and 6.1 of Polyfolds: A First and Second Look.

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Hi were right so I guess on Star saying that I'm trying to think in a way I'm trying to tread a fine line of In a sense that you know my main goal I think jointly with captured this week is is to sort of set up the analytic foundations that we need to make sense of what I was going talking about next week but and I and the thing is that I could do that completely abstractly because the the politcal theories is sort of completely abstract but it's also designed to be completely abstract which also contains useful problems so no I didn't really abstractly I think everyone will get lost frequently and not have it be founded by the way you wouldn't have much to connected to so I have to try and bring in sort of elements of things that are hopefully familiar and impair them so you know just the right way I think with with with certain ideas coming from the from the powerful dairy ought to try to lead why definitions on the way they are and and so yesterday at you know I think might have been sort of a good you know a good talk 1st Nova was 80 minutes with questions in our 60 minutes with no questions and so I think things that will allow him to be and are and so I just wanted to kind of live wired to hold to say yesterday and then and then build off of that today so while also so quickly that briefly but 1 week when I try to do yesterday so the 1st thing was that will we wanted to try proud tries a neighborhood of a no matter were not doing nothing so the home of a cureall really trying to build this big Indian space of function so we're trying to construct some sort of TransAsia down some sort of a charge for this and so so the 1st the 1st observation that we made was that this curriculum that basically yields us the right topology and this is really big ambient space I in which by which has this nice property that in fact are compact modular space Nunavut has you know you're doing with no curves through nose broken trajectories or what not this big Indian space has a apology in such a way that the new compact defied modular space is a subset of the the correct apology on your modular space the compact 5 modulus spaces induced from the Indian Space right and so freely madam was something something rather rather useful as we define it sort of like this Our has always upheld this gives us that it is if this gives us our neighborhood in the right space will maybe we can use it situated as a privatization but we can't because it's Infiniti the wonders all this information loss right solve wall what you do what I say OK still using sort of they introduced this this might it's going mad for NT renewing the however might like to
02:52
refer to it as I introduced this again it was some sort of unpleasant formula that had this nice property that when the when I parents right what I pair it with the with the plus going where the previewing map that suddenly for each fixed boggling parameter this map here became by objections and so that the ideas that will Howard and make use of that well that's useful because what we can do is we can define the set script all sitting inside the sub space here right With the property that well what qualified all those maps that when you when you I antifree little or will you might it's cool you get 0 and that's good because added what that means is that basically we restrict our Pringle amount or plot spooling out precisely to this at all it said that necessarily being a by Judge released injection by ejection with its image Bell that killed this problem I would no longer have this problems ones are happy to serve restrictor attention just 2 0 but all might be some weird sectors and I might say well you know so what what have we are what we actually gained and so there was this notion In this idea because in the end it's gonna work out but the idea as well maybe in some way this set also supports the SC calculus and that you know maybe there's some way 2 to
04:16
treat it as if it were something like a bomb workspace for something like the scale 1 space maybe it maybe you can define a notion of of of smooth match between the maybe you can do that so how might you do that well you have to do a little you know a bit manipulation here dictated terms to rewrite the problem and so if we did that by sort of writing function are again it's domain the same images into the target space anymore it's a map from sort of your domain of of free glued argued your domain of unglued but maps basically it together with good parameters back to itself again we we defined in this fall in the following way so we take we take good forever repair maps are we apply this box schooling map which Azerbaijan action we should say this here this is this was asked yesterday this is my rejection of the 1st factor
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I reserve 0 0 out that the 2nd factor at least in we take this box schooling and then we made we made
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some observations or over rapidly and try to make some observations as I would have spent more time with if I had the opportunity was at this that although we had a definition for right appear well once we take what we take this definition of all we see what its powers to find we see it is precisely the sort of fixed points of law there we also sought check really quickly that they are composed of itself was just Oregon at the projection basically the nominee projectionist that sees it it is easy to check your basically because this is about objection recent side projection between and adequate computationally eligible oneline proved basically shows you that that these 2 properties together guarantee that this at always in fact the image of all are going and then there's the experiment is theirs said after giving this space cross is suitable as a scale 1 of space structure this map are SC infinity I really I hadn't really give you any justification for that not community justification for that you can look at the literature I and you can sort of see Will why something like this true and if I remember correctly I the proof of this essentially boils down to well you know you you right out why we have exposed a formal offer for women's was a formal offer for all here will you write everything down you get some terrible sort of equations some in some some awful composition of revisions in products block you break it down at all little pieces in the end day it the other day prove each little pieces as he's smooth essentially while going to that list right but that is in life in the lecture notes for day 1 hour and that tells you where to look and as calculus book Odyssey calculus paper to tell you had approved each 1 of these components as he's moved so consequently you you rapidly built back up from this but this is this is not in fact as the smooth so that by itself that by itself doesn't really seem to buys anything short please do they have the will and the fact that he is not necessary that this was the place where the families of the price of the shares in the new alliances so I'm thinking of so I'm thinking of Caesar as an wears off or 2 which is you know a nice Bonner space of finite dimension so I can take it as a sign of the American raising the ceiling for Lowell a I mean it this was a suitable as really applying to even out the season was this was interviewed in the film and the most died here is list of that and I am not serious said no no source of unearthing I think it was a sort thing it had ended up being sort of a slight modification and don't know if and my lecture notes I actually defined scale wants base structure on the IDB something very close to it and I told the based apology on the and if you understand those 2 you can obviously guests with this restructuring is to be on the I don't know I explicitly which is why I'm saying some thank you whether any other questions that I know what a ball through this this is actually fairly important yes to the purpose of this right so yes I am so let me see I want see where this so here's how I would explained so so all is defined to be desirable points with the miners schooling is 0 right and we define it precisely this way so that the restriction the plus schooling ends up being a by objections so then when you do is you say OK well if you look at if you look at the maps right so so this was the sole scrap then is going to essentially be the set of all points which you which are going to are going to but parameterize your neighborhood of your sort of nonlocal maps plus some noble maps as well OK so what happens is if you think about anything that has no map we set up sort of a base problem you typically want that nodal point to say Sadler model it so it goes to say the origin and are 2 wins for instance right OK but these are sale can wear only nearby you know nobleman a Nomad of Oregon be sort of some of the ordinary knows or even nominal those larger nodal you 1 noted .period to move around so that allow this notion and I say OK well I have to allow some of no points to move around his forces you to add in the sort of this extra constancy but I had In the definition over here with this abuse of the legal correspondence cabinet space OK that's all fine you can do everything with the previewing there's no problem with the prevailing that but in the mine is going right or what happens if you don't have those eager not subtracting off those averaging terms then what happens is when you when you look at this here it turns out that without those averaging terms it necessarily must be the case that the only way for this equation the hold is if asymptotic houses a 0 and therefore that the the the node can't move around the image you think about it do you know who you are responsible for most of the rest of the nation on Monday morning when using real jewels this sure the other question all right so this is where we left off so remember I guess what we're trying to do is to try to investigate maybe crypto here supports as the counter strike that's sort of the idea and so then so that they really serves I added army and Idaho sort developed in practice but but but sort reading through the material then you make this sort of observation you well then that's the case the have map all are mapping essentially Warnock space to itself which is SEC move and or compose are equal are will that I could do this track right could that I could do this trick I could say that a function after flying from the image of all are due to the images some other on some other space right it is but as if and only if this is my definition if EFP composed are is SEK as a map from this opens on a scale of 1 of space to this opens up scalable space so we already have a calculus or defined as the bonnet spaces that we have that defined that now the ideas will have ourselves as well of the substance happened to live images of the special political retractions this is the case that we can make this definition and all prime right so it's so there is certain there is 1 good thing about us all the terrible age always the notation is it's fairly consistent and so are so it always ends up being that that all I always always the image of you and he was supposed to be a subset 8 an open until embattled 5 or 10 minutes In which case we add incidents additional portion where put in something like I think it's called partial Kony and the use of partial quadrant in between here in which case you was relatively open in a partial quarters but then there's always ends up being the domain of R & R and all are always goes from you to it has to go from you to it has to go from 1 side to itself absolutely necessary as you would want from a retraction any further questions OK so I make this so now the ideas we have is that we now have a definition for small functions between these weird images it is weird subsets and that this on the 1st bit of magic a 2nd bid is actually the strange sunsets and they really are and rather
14:30
strange right down to examples I think there's a hallmark example on this not where you can see these things might have finite dimensions but that fundamentally that they the dimension of the space might jump right hand locally varying dimensions lovely Renko dimensions might be sort of you know of full or it might deserve a full are said of sort it's it's it's a symbolic space for instance which fibers over some other finite dimensional space that might be sought a full dimension on 1 region and sort of have infinite could mention on another it's a very it's the the Opry or it's quite wild but nevertheless despite all the strangeness it does have a tangent bundle and you can just sort of see what it has to be and I have a map all are which which defines it which meant you to you and or compose are equals or are and so I say well then it must be the case of the tension map or maps to you to you and you apply the chain rule general Cezil Tr composed 2 yards to yacht which gives you would get a map precisely of this form right and so they he just take his definition the tangent to the subset is equal to the image under this the tension map of art of attention to you it's kind of fun corals are the only thing it could be and it's a good thing analysts the 2 were not having do I haven't officially defined scaled skills with attractions but this is sort of this sort the 1st observation so that so that once you've done this sort of on the almost stupid things and just tinkering you find this sort of stuff right what once you see that each then you should have this idea this is the big conclusion that should should be well let's try to build manifolds locally modeled on subsets like that so what are all characteristics of all that we need in order to actually build some sort of something like a manifold locally modeled on as this is this is what I would have liked to have conveyed I guess in my lecture last and I think it's important sort of see the story completely laid out like this on To see how so these things here and a big so it the case this is doable also this in just a 2nd but these things are called off a season would retracts these recalled as he's moved retractions and these essential reform that are a former local models for Empoli folds and why what what I'd like you to be able to see is that is that if you start with just Pringle Inc which is something that shows up in in whatever framework you want frightening away CAD Framework you you you had notable broken up elements modular space you start with that in some sort of classical analysis that you do this trick to introduce this might glowing repackage it into this into this weird sort of nonlinear projection and that says he's moved they necessarily have led to this idea of of having these Estes would retract Wichita provide local models for big Indian suspects right so any questions about this outline this is the view from the area of the world and that this is really different from the norm and he said that this is the procedure which the lessee infinity and most civilization ,comma 3 composition but yeah think this it is exactly what you and that is of this week this well but I mean any I think it right if I have any if I have as if I have a smooth if I have a bit of a classically a classically seek a function f and I pray composed by a smooth functioning me my general should tell me that I just have the the regularity in Athens should be preserved so again that's that's happening here so that sort of not so surprising I think the surprising thing is that the surprising thing is that if this was that this was commited yesterday on the surprising thing is that if you if you reduce this from SC infinity brother increases from as seen through the classically Infiniti then the image can can have this really weird properties sort of necessarily must be a bottom fall In the then reduces from the standard calculus and King yeah I thought it was the 1st thing in the morning on followers of the mosque there OK so that's it that's a great question I want to get to that right now actually so so there's this question about what I say what I say when I say locally modeled what actually constitutes a local model right and I keep saying Oh and the 1st thing you see this are the natural question has to say How do you do you don't need Oh you mean are in there somewhere like are in the cellar but sold its I definition really quick so the isn't as of space same could contained in the car is open and will recall my ambiguity irony in opening the based apology but all are as a map from you To itself satisfying Paula composed of our peoples R and R is there see infinity at all this says that is the definition the analysis smooth projection I then I'll say is another definition which is that of he isn't as advanced space on script subset of the it is an SC infinity redirect provided there exists In all which is an infinity retraction such that Oh is equal to the image you wonder or when Henry will sloppy but everything here is basically just me writing this stuff down in Canada the minister said that so far has
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he wants this kind of war How much of that structure is right soul OK so before OK so all point out that your 2nd question 1 answer your 1st question 1st and the 1st question what we mean by sort of low What is the local model and so that was the previous question so here's here's the answer to that question the local model are are the following 4 and if I want yeah yeah not not this and not say all you are you ,comma we don't want on here which is strange but it is also the case this is also adjusted terms of notation I can say that even recently have been highly irritated by the fact that this back in the UK that you can't just talk really honestly if you want really be honest as possible you can never just tell me always a local model if you really being honest you have to give me bowl all in the ambient space but it turns out that you don't have to tell me all are and the reason for that is that if you look at this definition but if you look at this definition you sort of say OK I suppose I have 1 retraction which makes this function here as he's moved right and I choose a different retraction which defines the same status then again it's going to have the same regularity sir regularity independent of his retraction and the attraction doesn't really tell you any other information so all the structure that all has all the differential structure I mean is essentially a mail the scale structure is induced from the and any differential structure is induced from the fact that there exists an are which will whose image is up as long as there 1 your fight and so in the larger than those that becomes a local model this is replacing I say open sets in or N or open sets in you know Bionics faces right and I can make that even more precise now lesser further questions by anyone the company also it stated yes it has a skater and the help of the yes so so all remember it has to be a subset of the is so any time you have a subset the natural the natural thing to do is to say that the case level here is just equal to all intersect if appropriate scale structure there and they know end the comment remember my ambiguity the yesterday Service says that says that says that when any data Madison WI statement like this is that I was in the basal but it went on and I the most the size of the business of Manuel for instance the only means that the majority of the crime the you go from here Oh I see well I'll be honest and say I don't quite know what you mean by isomorphism in this case I mean you know it doesn't it doesn't necessarily have the thinking necessarily has a linear in general doesn't have a linear structure to it so yes in general doesn't have a linear structure to it and so consequently isomorphism would mean diffeomorphisms and that is a concept of the but in fact I but in fact made from just what's on the border should be able to 2 conjecture what it would have to be you would say Well you know I've got to such retracts and we say there as diffeomorphisms de s Divya Moffat provided there exists a function which is about ejection between the 2 which is SEC mood RSC 1 whatever I guess what will stick to the smooth category I think for for simplicity well I mean he is sort of running the background because I can't talk I can't talk about I can't talk about a local model must have both of them I have to have the ambient space with the international community this is a list of goods of course are very this of course is is the actually to Facebook post had about whether or not it's also acceptable to write statements like this topographically most people hated statements like this but you know writing down the sort of writing down what the the subsets of his sort of useful life the vertically it's always slanted it's terrible how can I say the With surprising as Maria this is what said that we understand that this is In the wake of the question I think that the fire was this right and NSI said by injection and that I want to note the function and a divided state and its inverse Devos would want 1 direction sorry if I missed that here is widely used and they have a knowledge of the world last acknowledges that the the functions that's part of my great and I don't much In the 2nd half of the year the the U.S. simulated stake analysts refer to him at the commitment were worth and anything saying that land he said that here in the heart of this thing can you can get away with the suggests that the year online and I'm not entirely sure the question is isn't whether the the only thing you know who has not yet been here thank you I'm sorry that was my
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fault and the islands of remark look so now lesson and it is something which is sort of of this sort of obvious and I think it's obvious but I also think sometimes you're doing obvious things on the boards of solidifies obviously should be sold a wellworn do is is basically give you definition of it and colorful minus online so let banks the topological space the next in charge around next is to many fly ball so that this here it is opened in excess this is the best personal retract mean in the sense in the sense here on the local model of light always a retracted inside the and then fight is a map from the 2 0 which is a wholly amorphous but as of this point we have no further structural and acts the Commission said infinity and that was what they was misplaced X consists of a set of charges of falling forms of seen such that 2 definitions such that rooms they are terrorized compatible compatible slaughter mystifying and this collection of the the cover it backs 2 other "quotation mark was that named you need is the who sorry to lose and Nolie he need not be fixed so any chart of this form where he is some scaled 1 of space always some SC retraction retraction sitting inside that we are these any open said Pfizer so they could on different that's actually started yeah I mean things like that things like that are are actually important because you know when you want to consider maps from CIA domain is romance surface you can ask questions like What I Really Mean By Ramen surface right so might want us to be sort of you know this set of all points of unit distance away from near 0 win in our someone else might have some sort of slightly different but you know they want to be sitting inside our for something stupid right so you know you're domains are differences strictly speaking a different spaces but we won't allow that in in such a definition In assurance on compatible and the associated transition maps on the school the Miss this is because there is In the end soon and that was thank you definition important and and and call for news Bloomberg the company was misplaced equipped with an equivalent classes
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not claim this definition should be sort of obvious but also of sort of necessary for me to write down there's a question over here I think not because of the questions it's any questions about this I was well because I wondered what I would like it to be the case reversed at finite dimensions are from the usual notion of a different woman manifold we want what turns out to be the case that we you build that these are Empoli folds for instance for a growing awareness of tea and stuff perfect houses the standards were there right but say it will rightly Haverhill yeah Solis has so what happens and that's a good point actually write so so what's happening here is you starting with you starting with something that's nothing more than a pair ,comma Pericom backed up parent had has topological space it seems that no additional other structure has no level structure doesn't have there's nothing else right now and and and so it turns out is what this is the structure that you need so that once you equip it with you know article was classic out his room 1 Alice in particular because it's the case at all of these but all the associated transition matters have to be a seasonlow they have to preserve levels that have asserted differential structure any information that you see in the local model that he would like to see in the unpalatable but also preserving the transition after the decision everything we've discussed we can then claim is just induced on em the of the of the local models so yes it is the case that after you have analysis for this thing has a nice you know what I would call scale topology and bright mothers from is the Church of the From a subset of the substance of the crime not at all year so this means that there will be all of this was helmets exercises stated the ISO it turns out that indeed turns out that I did that did that the restriction of 2 overlapping not yet it's destruction of 2 retract also generally travel also open subset of retraction the yes that's a good point In other questions in the House the head the a wood can be we can't beat me in general it's certainly not to be open because to be open is sort of Philippine Open set inside EU its can much wilder but you know it is the image rendering of an SE retraction of all the image of an open sets His cabinet space by but by a scale retractions skills with retraction and assertive enough to make this definition you the whole thing welded to the mean all went strictly speaking as you know scales apologies with the base 1 yes is induced from the debased apology you yes I yeah OK so now I can tell you that I can also line are in the following sets so you know if you study Nova not studying role of women in pretty much any other should modular from the likely that to come crossed all going to have some more rich algebraic structure which are relies on having along realizing a modular space having boundaries and corners and currently the way we have this definition of Empoli folds and nothing has been undergoing corner so we need to fix that I "quotation mark an a the places the the creation of a time for the the it's now where we have had SES morphism is a linear SC 0 man Teamsters helmets and it the U.S. which is why so fond of levels partial finally towed toward those Here's a Quarter Interpol is closed set close Bloomberg said
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closed convex 718 I see such that C equals to applied to In teaser Ellen morphism so this I'm sure is a partial quadrant Wikinews partial points as well you know fantastic will squadron and as a model partial quarters and you might say Well why am I doing this I'm sorry yes just naturally justifying the arms a user while I do it is not what I wanted to is is what I really want is to have a notion of of of boundaries and corners sitting inside sitting inside a skilled 1 expertise and so on this is sort of your model for that the same way you write down you know your obvious candidates and even invite dimensions sort of have this shape where this is going to be to the end say for instance replaces the 1 expects that's fine but and and there which is what we we allow this advice morphism so we can move things around so you know in particular this region here is now partial quarters I'm sorry W's some other scale 1 expects but it was born you yes yes so that so the latest said over here somewhere in this notations are always different scenes last season some partial code in size and scale 1 of space and we want to be the image of the sort of model 1 under some of the nearest years morphism the cost of quarters you we wouldn't who in on it who's Oak is averse voices so that I'm not the box which is the corner of the box is allowed to have all 4 corners because it has to be because it has to be a little which they have been if following day the quiet and what sort of yeah you're probably not so for instance if I have a period with 4 sides not allowed right so as not to use you could but the current theory doesn't allow this the reason that I want say I was reinstated as nongeneric if I sort of think of my faces and move them you move them around sort generically than that nongeneric position this congress opening the Europe and the rest of the status of Jerusalem as long as Hong Kong In New York the course Yukon condition afterward reducing the use of the people .period yet violently so observation this Agadez extends to 1st quarter system with 2 different calculus extends manifold boundary in corners the same way sort not surprising statement as with the definitions and analogy and make a slight generalization so I can't be less dishonest and has seen fit to retract his words were replaced a previous definition of who is really in trouble 0 he previously we didn't have the seat here so that this is a scale space but this is a partial quadrant and this year is once again the image of a retracts but now you it is a subset of which is relatively open I mean it's opening the sub spaced apology by induced on C from B the sitting on new yes sorry and all is that as SC smooth retraction he
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said that next year the end of the day odds Lydia so it should be the case in inventories induced from us 0 so why bother with you all the things you want to standards all Standard quad renewing that's a good question for elements are I just think I think it's it's useful to have I think it's useful to have this definition at your disposal when you need it right because otherwise otherwise you know you might come across a collection of functions so that the waved find answers the easiest thing you have defined is such that rise I think it's convenient what the question is why have we bother to define partial quadrant this way why don't we just always work with the model partial quality so here we believe status of the notional brought forward at this point to go on the field but there why while we defined the instead of working with the model partial water here so far the board or the state court in this for all clubs yes it is naturally defined by something and well yeah yeah I'm not convinced either way it but I have some somewhere talk about this there was a lot of those at the site a mail order because of conveniently are voice spaces should be a finite dimensions and therefore than the the dimension of the boundaries in another the highest or a corner you could possibly have is finite so I think the government connections will always be it was the image of the whole text of the letter you go to the of for the moment the reasons for it From usually I mean you know in order to have an infinite dimensional corner you need have an infinite level building and in the northern eastern standard analysis which come in here and there the history of the sun right so what I would like to do is but this this is the only 1 of the names of all the hallmarks of the long the closest nothing more than a of the woman probably will potentially useful I write about what you say what you're saying is army but but what ends up happening is you're looking at like some big union of spaces these these sort of non notable curbs on and then I guess what I want here sort of some sort of a cross fiber product here right so no you be union the fire with itself the union be fiber products 3 times as he take has been 1 union and no point in here is ever a point in an infinite number of ways "quotation mark admitted that his goal the people the front and rear you could no 1 that has been in question goes all out of his way I know you all the time the India all over the United States has just come this sort of state's only for the more or more on the way home about 2 years we're to this is the story of the National Union of Poles almost cases this year I was going back to use the utility is based the whole thing that I want this time it is something we actually want the provisions of the function in the course of all this and we have such a focused what do would you think it's just inches of for the managers because of the International relations of the this was it was used stays in place so long as the government of the bridge the policy over the last descendants of the right which need has generously let me take about 10 minutes everything from Seoul let's see what I like to do signed minutes of my notes are more at my notes are more details on the right of Lord here but I'll drop on some some pictures just presenting ideas that so what you want when you want is a service OK if that is and then Polly fall of citizens with so OK so what I have organizations that is so once you want to have been part of the retracts all model were generally they just you run through the same collection definition in terms of charts and that was to have a new definition of and powerful which allows the your local models that's the honest definition of an animal rights and now we have finite boundary in corners but yes but it but it will be the case this in that Empoli fold you have sort of corners of whatever dimension you would like not infinite but find whatever you like OK so then so then suppose we have pumped up and unpalatable would like to have a sort of a measure of the corners and ensure correct I'm so low I mean by that so so right so we will really want to do is to find something in Helmand produces a lot of the generous index generosity index but which really what you want to be is a map to find but I think we just need elSayed DX is a map from the Empoli fold into the natural numbers but can serve 0 and so
54:52
what I would like to do is war how honey defined as it's not too difficult up with his services as a whole apples effectively defined through transition maps and local charts was justified for certain model case in the model case you know here's a nice you know here's a nice on on on as symbolic space and if you want to you know you've been put across the Plus which and others Kilmarnock space and these points and here the interior points this is where that generous index should be 0 and these points here are generously next 1 these are points of the genocide indexes 2 and so once you have in this sort of local model in the standard had asserted generalizes more distancing of generalizes Are you can then do a proportion quarters made that definition you then are you can then define it in a chart so you then when the notes for instance still see the genders index find like like this right this is something natural number right and then I and then from the charts you wanted to find it for and Pollitt Foley had uses this little trick here because presumably you can write down local models where you can write down local models for the a robbery Oregon identical moderate models for indigenous index is different depending upon the local models you take the minimum overall you possible charts and this allows you to serve essentially defined Jenna's index measuring sort to what what stratum of the corner are you in right and then after 1 defines this it's important to stay following proposition if expert and while I'm bowling balls St can't use contained in taxable these contained in y open said and that maps from you intervene isn't there seems have to be smooth the states moved to field then then the 2 generously index on acts of point X is equal to the Jersey index and wife of death of the next here were assuming that Texas isn't new careful X is the so the conclusion here is that as he stood diffeomorphisms on Hambali falls preserve the sort of corner stress which is exactly what you'd expect in finite dimensional manifolds with boundaries and corners the units for the truth is that started this this is the part of the review board yes we at call the Securities and Exchange Surveillance System which will also I had a shot at the end of each year and that was in the movie he said that you didn't have to hit the coasts of the group which produces of I used if you want to the licensee only reason that position last year the bad batteries that the minister of this this and this and some of the loophole also if so the point is that you might have you might have some stuff like this and you don't have a retraction of sitting inside here on Canadian and 1 yet so tried sitting inside here and then the image of other retract in this season's twodimensional case might be the sort of life and then if you have this line is a wall you know what's the Jersey index here what do should be 0 of course was the generous index here well is deep also Wanderers equal to 2 In this model is in the ambient space where it should be too but if you take a different if you take a different local model you can find 1 words 1 when minimum questions but you know you have here so I said of you made it obvious that spaces as these signals media was also there but it is what it wanted in the United the 1 How do you know that I can't make some sense something like an Empoli fold up with a boundary in corners making use only of this definition which doesn't allow for the development cost structure In 1980 this you have the right to generate interest rates going so known as you cannot be sure of that you who are committed to overhauling and the he said that there at the end of the year we started late so I'm only 2 minutes or based on when it started but yes I can I can go ahead and start here will you yes I guess that the last the last survey I want state about state about this is that on fire dimensional manifolds with boundary and corner that recorders have this sort of nice property that on if you look so so here's another toy example right his use toy example they have this nice property that if you pick up the you did along the boundary the of your fine manifold boundary corners you can do so the following things you can look at set of all points of the generously index 1 and then you go to the closure of the West the closure of this is again a manifold with boundary corner structure and so I can see that this picture that's just sort of this sort of here and that's a property that finite dimensional manifolds about recorder have you'd like to lift that up to Empoli falls as well and that can be done but there's just there's an additional condition that 1 used to place a new definition of attractions but they did define your local models so this is and this is the cold attaining condition it's in the lecture notes an octave presented here and the key thing to know is that the conclusion guarantees for you this this nice property you with the closure of the of the of the fate of the faces of the New Jersey Nets 1 portions and these again Empoli folds barren corner that's 1 and the other thing to remember is that essentially all retractions that in practice are the special type the sort of splicing type attractions and these are in these cases is that the additional obtaining condition is automatically satisfied so it really into being something you don't have to check too much I mean it should just be should be completely straightforward and then you had this additional profit OK enough well finished I think going urinal few but the people here like since done already know that the lecturers or online I know read them in your leisure and of a yet of course we have the questions you can ask and I will be speaking briefly in its discussion and I'm sorry for that but I have to make sure enough stuff is done for catching to start tomorrow so the Saskatchewan In the questions raised in the air the union most are justified in Rafah as the on 41 and powerful if there is a good thing at the end of the Cold War and Peace and and you know that in the hole this is the sort of it was the Sunday was very generous among the top 4 fewer additional minutes or just to finish up the stuff and I want to make sure I presented before catching tomorrow is my voice is just about to give out I think that's the universe saying it's really time for to start talking so I'll try to be as that we could as possible so the main things AllStar over here strangely that's the sort how things that laid out on circles 1 2 3 4 5 so I'm good the names of the main thing I would tell you about is the statement of the sphere sold where were spoken most is so far as Empoli folds which is said this is like a generalization about on a manifold and it's it's it's a generalization such a way that it's going to have boundary in corners it's it has are both but topologically and differentially it has enough structure but it has enough structure so that it will contain the compact 5 July space as a subset with rollout apology just induced from the ambiance of steel topology or based apologies great so now you have manifolds but of course if you wanna do you know if you wanna studies at all for curves say for instance wrote modular problems we want use to arise as as 0 sets of anomaly a from WalMart not only from Home section and so that means we need some notion of a bundle and so the main theorem that I wanted to point out here is this is this implicit function here this is sort of the 1st step right which basically says what will we do expect a finite dimensional version to say on the fundamental versions say without our manifold repairing corners and say you have a dimensional manifold I've got a nice with bundle the top that I have on my stick a section about bundle in assuming that the 0 7 is is up the legislation of the of the principal part of that section is subjective but they should be the case you have implicit function during which guarantees the 0 citizen factor manifold and that if you're and it is the basis in of the bundle a manifold with foundering corner then you'd like it to be the case that the that the the corresponding solutions Center 0 set is also has boundary in court and so that's essentially this theorem says on top of this board down to this line here right that's just essentially the full version yes there 1 on no so was so why is is the bundle in an X Little Axe little Wise said why is this is this is cabinets and I could remain this wall or of is that the issue this said the right thing the 2nd of the season dates with every point there .period 61 and the people thank you right so so that that finite dimensional version will you sort of expect from him the sort of imposes function their music is exactly what the stated says but there are underlined all words that at this point in the in the course of watches we don't know yet so I wanted to kind of go over those someone quickly that can over quickly and some of them I can't so so for instance a big 1 hears as seafront home sections of this requires an entire lecture but that's the point of tomorrow's lecture by catcher shall be introducing as seafront home sections but and then there's attained strong model which also takes a minute and that's essentially takes up the bottom 2 boards here so applause on that for just the 2nd and the remaining 2 pieces or good position and I worries about subprime colorful so let me go ahead drivers top or gender selection .period these years so good position has a precise meaning it is in the lecture notes as are essentially these pictures here on the point is that and I think the point is that in order to guarantee that the guarantee that year the submanifolds you get this sort of cut out has a nice boundary corner structure than than the behavior of that submanifolds off before insurer has my stomach or structure needs to behave interact nicely with the boundary of sort of the Indian Space the band recorded the Indian space and so general position would be sort of somehow it's it intersects transversely and that sort of ideal what you would expect I should probably mention this about music sort of terminology so when you read when they say general I think generic right the general case I in my opinion covers all possible cases with the generic What the 1 happens most often but not necessarily always and then what a studies he said good what they really mean is good enough you I right because in general I would say that you know I would say this position is better than double whatever I've drawn right here which will explain 2nd but I mean good enough because it's good enough to get the result that you want which is that news service of manifold has nice boundary in corners on my stomach corner structure are induced from the engine from the Andean boundary corner structure on the unpolished folds for instance right so that's terminology and so alright to here's general position and so was good position meanwhile basically what it means is that you have this you have a tangent and in it's a little trickier because in general of course have drawn everything with her some space is being like onedimensional and so you so it's a little bit trickier sort of make the statements especially since a adding spaces not finite dimensional on but the point is that you wanted to be the case that you have your your your your attention planes was gonna pass through the origin a wanted to be the case you can sort of wallboard a little bit right takes on open neighborhood of nearby planes and then you want to make sure that that open neighborhoods stays or the corresponding race if you will serve stays on the interior of year of your ambient on partial quarters and so that's why would you have this line right here is supposed to represent the despite international substation is good because I can wobble it right about the passing everything still through the origin and all those corresponding plays past the interior here that would be an ODI that not the case here this is not good case where you have this raises travels precisely along the boundary is now a wobble a little bit and suddenly it's something it's presently outside partial quarters here is it's not a huge national land of the rest of your view of the international alliances and the long years yes but the district I think he said no reason was that that was meant to be coming along absolutely absolutely and of course here's something particularly not good but when you have sort of a line that sort of an apparently this happens theory sometimes it has to be dealt with all you have some line which passes like this just disappointed and concerned season you know no provisions of this alliance would bring you into the interior so this is just another picture definition the precise definitions in the notes on but I wanted to make sure they're always be aware of something like that on a assault powerful so I can read the letter X B amp olive sought forgiveness and Indian space as a subset in there and we want to know what is a assault and Polly fold right while basking case if for every point in a you can find all the neighborhood around as ended as Infiniti retraction or are taking that open neighborhood to itself so the image of that is a intersect with neighborhood I mean telecommunications costs adjusts retracted them is much easier us so all right so I think if you think about this definition for a for just a moment you should hopefully convince yourself that this is the only possible definition that it could be any view in some sense replaced X with like St Bernard space and that a new short essentially get and nearly get the definition of retraction animal model so all really doing is saying I also have attractions in public polls and the retractions and Empoli falls can cut out some of Empoli falls and that's good because this there sort of tells you that subset that you get yearend holiday fold are you getting you in your Empoli fall is a solid and Polly fold in the bottom end of the service guarantees that that saw them Polly fold essentially just a finite dimensional manifold with them in court any questions about this the attraction was the only in the what right on saw them powerful it because it's in because it's it's it's because of the definition of what it must be the case you can find a local model in which it's actually an unpolished fold as well directions to find right but his should see that as long as you have something which supports the SE calculus as long as you have a domain which supports the calculus then you should be able to have a notion of of a scale retraction right so it's just onestep generalization from good so the last thing that I haven't explained and other is the notion of attained strong bundle this team is a market Uniroyal talk about too much although it is it's discussing the lecture notes on what we basically just saying this is sort of additional conditions on your and Polly falls to make sure they have a nice boundary on a tight structure so what I do want to talk about those this term strong bundle which the 1st Tennessee might see a little bit strange and so I wanted to I wanted to sort of make you aware of something so that when you see the definition it doesn't seem like What the heck is going on here right so so and to do that I could use some stuff that hopefully we all recall just from the classical theory so the classical theories says you know on a linear inches of linear from home operator is is that it's a linear map between spaces with closed image fiery dimensional colonel in Coker so this is fine and is the case the frontal operators a stable under Compaq perturbation so have said what I mean by this you can you add on contact perturbations in you still stay from home and it doesn't change the and index so that if you have a .period classically smooth nonlinear map right then we say this is from home provided it's the case that you when you look at the linear nation of this map at any given point lean years Asian is fragile so again classical as so that so that we come down to us to this point you say well was awarded that domains sort of too much on 1 and they don't have topology my domains on trying keep someone as simple as possible and in that case we see whether coach everyone operator is Fran home as a map from something which takes maps regularity Kate plus 1 2 maps of regularity cat naps is sections bond pullback bundles and so forth but the main thing I want to focus on is the fact that this is of regularity cables 1 down to regularity can sort out if we see that and that's true for certain any that we choose the natural choice to try and put this that this into the into the SC structure tied calculus so that even in 20 cases than for instance we think the Commission we an operator and acting from 1 scale space to another where this is the 1st skills of skills base the case level is has regularity K plus 1 and the and the kids level in the target has regularly attend justice the statement here translated into Farsi calculus language so it's not surprising but then something a little bit strange happens which is well water are Compaq perturbations that so we had right here in the classical sense and so now he says while we were met with we remember from from scale bionic spaces that the higher levels compactly and then into the lower levels this was a consequence of that invertebrate turnover compact operator were expected to be the case that it's going to be a map from ah it's good enough for once the scale space to the other skills bonnets space except we shifted by 1 some moves 1 level up if you write this down we look back at will look back in this case alleges means were tacking on a perturbation which maps HK plus 1 2 HK plus 1 right to have a differential operator which dropped to down a contractor basis adding a lower order terms so that's not so that's really not so surprising on but then when we what we point out 1 last sort of complication just in terms of and this is really sort of in some sense it's a notation complication they just have to push through carry carry carry through with it is the fact that well the reality is that they are applications the cold Gujarat's rate isn't just that it's not just a map from are you know it's not just a matter between scale bionic spaces it's been you really have to think of it as a section as a consequently we have to figure is the section from from locally at least from antic EPlus after this is our total bundle on end and so the court and and and if this has regularity HK plus 1 then we have to make them this has have regularity are age which is essentially what I've written right here and they had a perturbation where perturbation sort moves you want up the service lifts you 1 up in the target here and so I a written that down here so you your sections had to go from AEK to the Cape was R F 1 plus cash and this is sort of frustrating it's frustrating because that means it will involve it's no let local chart it's relevant you say OK this is easy enough to do the problem is the and then have to build a bundle and the thing is that what that means is you you have you you essentially have a bundle with 2 different structures you have this structure that you have to keep track of and you have the structure of the you have to keep track of and so the fact that you essentially have to using the factors that led to scale structures that you have to keep track of in order to have a good notion of perturbations together with differential operator motivates the words strong and strong bundle and so anywhere in the literature initial busy literature at any time using the word strong that means that you have to scales apologies to scale structures that have to pay attention to essentially because you're dealing with bundles that does not make sense damn it so this is 2 years older animals Compaq you as head of the the sitting at the head of the government had not only that but the issue of the bombs used in the war but only after all so it a good year helmet he said are it sounds so so I just sort of highlight something about the pollen theory is that on 1 hand it's built on this totally abstract level and on the other hand it we make sure it contains all the fear that we have added that all the fear is that we wanted to the downside is that started in practice when you actually write things down you probably could get away with not having to keep track of these 2 structures because they would be fairly apparent right but in order to abstract 5 had abstract theory in which everything fits you need to have this notion of having 2 different scale structures essentially on your on year on your bundle please take the hit now is the time when structure on level ground strongest suggestion this thing I know I this was the 1st time that the that 2 things the position of this is the oldest in the world for instance I mean you know that for instance it doesn't make sense but you might serve say well maybe you're lucky you got a scale structure here scale structure here so maybe instead of taking the usual sort of diagonal filtration we take a double filtration keeping track of both simultaneously with any right down a transition that you write in a transition mapping see me get some of that but not all of them right is there's a sort of unfortunate there's a sort of unfortunate fact that you know if I'm considering if I consider a map of regularity classical regularity CK I you know it doesn't make sense to talk about it doesn't make sense to talk about some of vector field along that maps of regularity K plus 2 hours right and you can't that doesn't make sense that doesn't hold in transition maps and so then consequently a transition that you don't keep track of the double filtration and if you look at what at what's the best you can possibly do applications the best you can possibly do is essential to keep track in applications is you know in general is to keep track of precisely these 2 filtration and from this you can get a partial double filtration which is the way of age too easy too so in the end what ends up happening is you end up reading local you end up with local charts the sort of written like this where we use the sort of love triangle here to the fact that this well it's so easy will tell you that you have a double filtration OK where there and is and forget this is afraid helmet the head of the costs and yes so you get this sort of double filtration but doesn't extend all the way up again the reason is essential because it doesn't make sense to talk about you know vector fields all of which have much higher regularity than the bed that the map along which there defined that's essentially what this with this but this problem boils down to but never but also simultaneously needing to have a of canonical class of Compaq perturbations in which to do the theory yes this is a list of functions right at this page strong bond conditions in conditions so that you have hold yes I always have like to have a hurricane is always prepared so that it can be so Ford Ford today yes tomorrow no tomorrow you will need to have a perturbation and at the point of me presenting this is that I have to tell you what the strong bundle is precisely so that all that will precise society have a space in which you can make perturbations of civilian precisely made you can you can make reservations for this statement I suppose I could remove but yet for for this statement I think it probably could remove all this are the strong condition and work with something sort of less than you probably worked through the implicit function Germanstyle clearly where dozens of profit fall so 1 this is the 1st original work the institute we want to do this was is a job that's right OK yes so inherent into the definition of a seafront home section is the notion of a strong bond it's it's sort of it's inherently built into it at which you you haven't seen definition so I've just been vague good but at the moment but I mean I guess the 1 thing that I kind of wanted to clarify here which I am does seem I thought I was successful at is is that when you see strong bundles they're going to have this weird new symbol were you thinking of sort of this has been a total space this is the total space over the base he and everywhere you see strong essential because you have to keep track of 2 different as structures and that and the need for having these 2 different as the structures is the fact that you have you have both of the frontal section any need to have a Compaq perturbation that hopefully I can get that message across if nothing else well but on because of the more the constraints of the stability of the Hong Kong referring to the the language of the rest of the world the since only foreign ones since the government and people who want to serve the interests of the peoples of Europe 1st and then the rest of the nation of it was but if you have subjectivity in English In the rest of the world what's going on and the people of good will 2 and when you have 1 the 1 of the nation's most is 1 of the reasons why that fish to cover the traditions of the past and I think that this was essentially a question earlier reduces the and I think the answer is that you know if you write if you write everything down in terms of the applications that you want to study the problem becomes much more concrete lies at the heart of the of the whole world of show you how head I would like to see him and then there is the final say on the occasion of the monument over the heads of cables along the lines of what it doesn't have anything this is the case then so the the rest of the world someone OK so I went longer than I wanted to and I really want and things originates on I'm sorry for