2/2 Regularization theorem for Fredholm sections of Mpolyfold bundles
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Title 
2/2 Regularization theorem for Fredholm sections of Mpolyfold bundles

Title of Series  
Number of Parts 
36

Author 

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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Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

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Subject Area  
Abstract 
This lecture will state a rigorous version of this theorem, and explain the notion of a (sc)Fredholm section. [related literature: Sections 6.2 and 6.3 of Polyfolds: A First and Second Look.

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so tool sometimes and yes and things for all the fabulous questions but I do need to get to the point landing so that you can take off next week so I would ask that if you have questions that technical technical academic academic you asked me after the talks so which means want to know what you should have about how in the world was applied to my favorite you will you for so why don't you just is of bright geometry questions of good for coffee Hayes could go His comments about the goal is to actually explain every single word so we want to regularize homogenized space which we think of as being Compaq said of some section we by now out what an Empoli for we know what essentially was scaled down from going the need to tell you what phenomenal what section right you really want to model things on the field spaces that we want kind of functions but I'm not gonna write fact but we don't want to worry about that right now so something is nonempty and what is not so it's space of other sections that are as loss something which going to do so if you think the plants would you should think Compaq perturbations trends general position to boundary if there is boundary and I'm going to be allow myself to fix and neighbor of zeros and ended all and I'm allowing myself to force the section the perturbation only happen in that neighborhood and to be bounded by mountains likens ski I'm going to be able to scale Mount down taken over 1 year so this pretty much means I can make my divisions as small as 1 and that is the reason that I think now with too riders think you don't need any kind of genera satirical needed in the statute make you could say that as close to to the unperturbed as you want there is also the Obamacare the writer if you like your solution in keeping solutions so what I need to explain that some was also 1 of the areas along the border home of the plants this right so what I mean by by controlling compact so if you just added perturbation in this and dimensional said it's totally not clear about this think which we're going to want to hear it's going to be some evidence cut out trans but why is the compact we're just going to build this into the definition and somebody has to improve that there are homes and neighborhoods that control compactness so
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controlling compact means that if I have any and the last section that satisfies these 2 things are not necessarily transfers solitude in the center of regular uses compact check what's what my wife ran right by yes why 1 quality 1 fibers and I should say that again at some point it's happen so I know the facts the image already told us about gives cuts my movement and asking for general position actually know exactly what my boundary corner struck off that's important if you had to flee the area was something you do a onedimensional money space you regularized and then you going get him some of boundary to this their because the 1 manifold now you like to know how those actually the ones broken trajectories and not the Tripoli broken trajectories I think so the actual boundary would be generous 1 it might have to this regular elections themselves saying that that's actually exactly where my to 0 said hits the degenerate index 1 local in Miami in Spain and in the space of 2 generously exactly come from a number of breaking to the only boundaries actually has application write to you might ask why in the world we talk about good enough so gun position something that happens when you trying to force coherence of perturbations you might not be able to make general position but that's because you've already prescribed the section on boundaries which is something I haven't done that's what good position comes from what started to but so in order to regularize now I should say that somehow whatever count again here is invariant under Johnson and so I need to do particularly well I'm really honored need to think about what happens when a very jail whatever In set up but only in the ceremony to think what it would have been devoted to different perturbations and the claim wall so also inspired most of these conditions except for transfers of reasonable economy except so I can make 1 from famine and then I will go on from a family again a little bit to get France with Ali so painting goes from PC to P 1 and I built into French section with 0 1 time sex so the need thing is :colon Alfred don't really have to sort of go and prove again that this is from I'm doing here is a 1 dimension in the domain that should you and 1 to my friend from index and I'm adding a new smooth section also compact that nearby so that if there should be freedom the stability of the should change index and all and so you basic facts of life to pull
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the falls this is going to be pulling fullfrontal it's going to be transfers after I wiggle enough by using them again and then I get a smooth what is it about which goes exactly what there was no longer the father and the only boundary have here comes from the interval and so Benedict reported from whatever happens and whatever happens once I wanted to write
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a box except that somehow statement that there no it's
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harder than proving that yes well and then to wiggle a little to get transfers of go so any questions about the state on academic work but I wonder what patients want a strong haven and colorful bundling actually isn't particularly to fix notations connected a waterfront home section but I'm going to just tell you about the special case so far suffice it all In all applications so I'm going to say L and the nice thing is that this special case will be things that are automatic strong contains not just generally for a fraction of the blessing of police for by splicing time it was it's not something that should be a bundle some kind of suggestions Polly folds that are modeled on supplies things went know but it's going to become a once right down the local trivialization so To get so likely to be backed up by real that could bundle structure on each floor space structure to be wise exit script if you were wondering if you put tell you might have to actually go into different that's totally alive but doesn't change when I change my perturbations it just might change from actual fire and I local positions at sea the exercise is riding the pony fall book to really take a differential geometry book and just right back to you .period fold from the roof of the sites at some point you to "quotation mark so splicing and only 4 models and now for a bundle I want bundle splicing so what is that what happens over evidence that in the base I would like the local bundle I need to be SC diffeomorphisms the new and fibers 2 as certain model Our again retract budgets specifically I'm going at some point probably forget specify all substance retracts splicing so I do that you can fill them and I you can just ignore this so so whatever this redirect that's going to have a very nice structure so there's going to be 1 parameter parameter for families of projections which give you base and the 5 so stressful things online my parameter is allowed to chew that's what comes from this is where the and final dimension parameter that's what the ban comes and then firmly projections on the 1 on the take projections which means I don't have to worry about skills movements of them by themselves just to worry about skills with the respect fact to that parameter and parameters right now I don't have to write once because they don't know what more pirate logic .period 2 different families on this what would be a retraction action attraction dropped so the parameters of the year the retraction just go to projection can apply projections this is not about to think and I would like to be projected on the wise use you have to wonder if that because that's what you have if you think about what me out again once given you the real examples I think the toy example 1 thing the fact that the nice thing .period all we witness now comes from jobs and manager of the fight I don't these images of projections so 5 years I think that's what I'm asking to be skills is I'm not asking Hi operated to depend continuously on the it will not in fact qualified but then everything is boring so if if there is to do with the then so if the operator topology is contending that the dimension of the fibers doesn't but in dimensions you can let the dimension of the 5 changed was exactly the example that it worked out yesterday and so you have this sort of bundled with varying dimensions of and I really want see these retract that is space that's where you have the section just like the current structure of the section happen over the sort of small little things and you don't have a canonical extension to anything bigger so that the bank staff here the that of ambient space is just exist locally and nobody says that they fit together in a nice way so I don't usually have an extension of the section to the please stop this is not what I'm talking about here and is not just the restriction of some section about half what terms we had some went to the local financial ambient space just local so I the should students and that not even say so right from the and the fire so we buy
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what I should say 5 images and then at so really great so this is the base that's where boundary and corners happened at 4 5 and the knowledge any ruling parameters just some parameters given the boundary and some don't the choice of words but I'm not even going to say why the sustained but it's automatically strong because he is 1 so strongly 1 of the quality fiber and in this case the fire given by Fred Zuckerman local model and I take every day In the bar but in the 5 I only take the fiber of quality want 1 so all right time example period so you could call this is an ample of Polly fold so what you see here is an old 1 ball that I've attached to the twodimensional sudden ring too again with open boundaries here and then attached 1 intervals so that a onedimensional twodimensional and threedimensional pieces of my ex and audits for him transfer section to come out something new here it's what I would like to be able to say and freedom makes 1 section over this that goes into this twodimensional demands smoothly and maybe even into the threedimensional domain 2 kinds of something it's gold to market . 4 action 4 4 so that is to be the case better have fibers of different dimensions the better job said about the same rate the base jumps and that's nice to explain what the splicing fight he why Over explained if I was 1 mentioned here should have said it note that space without everything the Points of Light Texture 1 .period 5 and so rarely at some point the condition is going to be really in this bundles splicing fibers jump at the same rate With that into a definition of freedom operator right now right this could be the cell to think that this job said that Johnson 1 right but would not be used who wide right and Maryjane how inmates example gave you a little bit of transition that are little bit of a charge for a peace here and then you might have to might wanna put new another shock just twodimensional minute you have of have an overlap but it just goes from this twodimensional piece that twodimensional PC you vendors some help the definition of skills to say in what sense that scares me what I do right now I do want that jumps at the same time the actions that we write this down and he shot I would like to so much maps it's going to the Fairmont fake to some open up this and then splicing now right topology which the thing of that's we built up and what is the right here this ambiance splicing was the union also are in this direction here In each 5 here was some line L 2 the funny thing is just that these lines all too well not constantly sort of turn into all the intimate dimensions of because the From Obama functions that somehow we could just 2 0 just gets pushed out it gets closer to the point so here so what I'm actually to take but some of the projection to be To the same little Rock just the 2 of let's go to the end Obama centered that's something like this 1 of these when the it's positive so but so far far they just here to the image of the little guys because that's just my base but now my final over there is going to be the same thing except that uses the same base growing promotor again so you over this I'm going to have the same fibers and he went every point the fire right now it's the same thing and maybe I should stay here right off the fire based splicing is 0 5 0 my by splicing false economies and at a point here 5 is onedimensional and so that .period he find it is also the ah local trivialization of the bundle why you simply going to be what began open so that what is that I like it points the ,comma X of that lie in thank you open subset In each of the 5 is again please what about rights 1 of the interesting point so that I find my blessings soldiers 0 and so far of my bundles of just going to be there it's a start because the fiberbased Johnson going to have fiber of the
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final of the whenever the fibers also jumped so that when dimension jumped nearly equally in some of my friend and exactly stays constant which is a good thing it's not building and it's going to be built into the the definition of a nation that anyone 1 could say so let's actually do here for a while they've found fault these I think the pride he has failed to the catch that's a compliment and we're going to have to fill at some point strong me nite bond by 1 is defined which in this case is that of automatic defined because we have knife so we just take the fibers of quality wines in the In the ambiance the property of general Empoli for bond is automatic for Empoli for bonds of splicing and what you want to think is when you have the right with but the real examples that it is time I think this is what do I mean by this might be a good thing In a sense I'm going to need it the Just trust me this was going to be built into the the definition of federal building great yes complaint against you out of all of the right to complain but it's the way you should really think about the strong and why 1 is because fibers age 3 over age 3 maps we really want to think about strong 3 in the pullback tangent bottle makes sense when you was in H 3 the usually all find over a point of quality H 3 is usually just age 2 better quality while it makes sense and it's sort of a bearing on corn and changes except right survive wanted to make sense of age fall over an age in which 3 people bomb probably go independence and choices of the trivialization but the nature of the it's going to be independent Aaron the best of hard to the really at it's going to be very sloppy so I want to think about the bundle nearer K followed broken because I'm lazy and I don't wanna say as if to you could think have fold S at a building like this a careful broken verdict is the same thing as a K plus 1 floor building because if you want break k times indeed OK plus 1 trajectory and the prospect of going to be totally something that of course trajectories tomato by are at least and I'm not going to do that but but I parameters right now so for every floor then when I go to get anyone doing from exactly the caves and while they watched the hot because we come from interior so every actually breaking every brief breaking that's what gives me the boundary and yes yeah and that was just going to give me my Internet and could not be seen here the I mean minister to say I'm so what if my bundles splicing here so I'm going to keep going from a needs and based space and some fiber space and I'm just going to tell you what the projections on but I should tell you what it is the space right the want to parameterize the trajectories and very severely for every trajectory it however some section of a pullback tangent .period and then it was a very eminent the flu the operator could to push here operated too I should end up in the fire so far more than just saying aged 2 and the 5 because I'm thinking about an operator of order 1 this and then right I'm going to be late for now and just right at the coast lies in the city of to tell you what the projections actually are come right so really and each of these is a right escape to bond is OK because 1 2 polls so quiet so long as the the projection retract draw defined at length so it should be the projection tour please kernel of an antigay along with kernel of the class :colon who rates so whenever I see variations of the 1st trajectories really I'm thinking OK very and I do it together that's my job now but that has ambiguity so what I want do is I want that have reduced the ambiguity that's when I go to the kernel of the undergoing but by reducing the ambiguity should that somehow not to what the point actually is in excess of heaven mind that's why go along the the don't actually want to change the point that I have in mind right now you're going to see OK evidently going problem is missing I think it's just worthwhile to say this again so what on TV or previewing they right of brain sections and then I think probably the lot of fuzziness here rights have to ask yourself can I actually the and then I should do this with some parameters and I'd like to be somehow so there's
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always going to be some growing profile in the picture that takes small parameters 2 and the choice of going for a profile of that that is 1 of the foundational choices in this whole subject is pretty much what is 1 for which you get smoothness and that's unfortunately not sort of obvious when you choose from 2 different Mumford space eventually somebody needs to prove that the variance that we get out of the independent of the chosen growing profile yeah Universal so I decided to something work but then there is the question whether there is actually a total order under control and the death the 2 of you know about the change in the way of the schools the year is using a he possibly we don't know that you know the announcement of the meeting of the Cold War and the creation of a section this was a resident of the wasn't normal stuffed into but let's keep that for next week right so what happens right so strictly speaking told you how to bring on the base but you you can do exactly the same thing in the fiber for the fire then again go goes back to some of the vexed question of what do I actually have a section off L I don't have a section that's just lives over the broken trajectories also In called my section lives over unbroken for the trajectories it and that takes an ogre movement rejected applies a further operator to it's the just 1 section of 1 Of the something that lives over a blue and so in the 5 I can and need to have the same demands that I have this group demonstrated the cylinders so I need the same domain and the fiber that I have in the the license do Mumford parameter maybe you could also think of excellence here on the splicing the parameters of the deal parameters and they effect the maps and sort of 0 1 forms in the same way the wife of 1 of the things I couldn't find it on the space except has nothing to do with it I write maybe right down track see what we I have used my idea right so what they might do bar operator really previewing and things really what to of take takes a big 14 could tool Dave after the free right on most of its range it's just 1 D bars it's not broken the opening of the main Estrada X's unbroken once this is the main stratum of my operator except when I hear broken 1 this sort of want to pull all of this back 2 something that should be in the final it's something that should be in the back writes some of it comes from a choice going parameters all these trajectories that I go together to from advisers .period and apply the Debora operator and then I need to go back and so I can take them but it was what I'm doing here right and going to the Final to get to the final I to plus minus office exactly and for minors going to be easy To this it's
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fun it that way I can this way pulling back bar operators just on singles cylinder pulled it back to this sort of to pull of cylinders the price of oil applying and instead of undoing the however as you can see right through the front operator losing a lot of dimension here it's massive and this is and then I go back and have a massive well this is never going to be subjected to a lot of things that don't match infinite Colonel here an infinite Corcoran here and so I think it's going to be too cancel the the kernel here by the Corps I'm not sure I'm going to get but that was my real life example and again right so here's why 1 now comes from the local things that are the and then the ad that actually age not just age 2 so that's really what I should be thinking of that but the 1 1 right that would come to this month but what filling so as again once and for all let me just quickly work what can I say now but so mean down my laundry list actually right needed to tell you what an exhilarating moments so what this is is a continuous function on this betterquality on that's a complete change by classically can you and in the example well you take the moment added .period what does this mean you have to be there at 3 the boys and now I'm that's probably an exponential wait on him because of what this to to be a scale space cited in their own and is supposed to be 1 answer be Compaq in the world except for conditions on the Net here you have you have to know that there's something really good small print the right side of the aisle can see all the small print in the form of it's in papers and interviews trying to find where about specific small difference would recommend looking at what we used to call the guy .period at a glance at the we also posted at on the page is lacking small price however always precise besides the place they publications was conceived only for the 2nd day of the became too long In the last section but 1st of all sections something that gives you the identity when projected down the plan if it actually takes values in the better quality fibers and and section off the bill "quotation mark Bond which is exactly you on what you should think upon contact operator compactly supported would be released I the day that the classical perturbations French J this something like Jeremiah minus J prime duty that is evidently 1st order of business what if the so when not what happens in the regularization however Mohammed copy of Jay's
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fits into this school board argument that makes change of jet built into the area after all what is it some particularly the needs to take a major threat to 3 right now I can right now I think we ought to say what it's I'm the thing right but know what that means and I explain what controlling compactness Mintz said now this is a list of places where Grand pulled up in in a basement would have a little bit of trouble making up the whole theory so far quite obvious 1 I need as I needed to be regularizing which means that if I'm in the quality if my section takes values and the quality of care 5 that I actually want to have the base also quality OK which is like saying Give DeBoer who is in you also known as elliptic regularity that some sort of the troops will operate as we usually look at and then the key point is this feeling that we've alluded to and as the last count the filling somehow strangely not just about solutions but everything that's something sense of quality infinity so New .period Ogletree and and I'm going to say what the the action on this front however is going to be easier to define and that because this from happening blessings and retract and this problem is going to be a map between on day of maps right between Bonn spaces spaces the space good news for the moment undefined and tell you what the filled section so they can so in the local shops I'm going to respond splicing the it's going to forget opened want right on so that's that's in splicing upsets in here and this bundle splicing sits in the bundle splicing accepts in and right so what I would really like to do and I would like to fill up my section here so that any chance of this being from for that's what could I do it's so 1st of all my problem is that this is not this view is not necessarily in all but I can down with my retraction and then apply which sets in he said that's not so bad but let me I should say what his ass actually the new home of for 1 of the rights of the section takes me Of the interview and then something happens in the final just gonna write whatever style of 5 and so that they can apply after the fight on so I need to throw in a projections like but then I'm ever going to hit the displacing within this never going to hit the the compliment someone asked and I'm gonna let whatever happens just depend on the complement of each I can write each 1 minor .period he and I'm going to use the same splitting and this is the last thing the complementary splicing go ahead but this guy
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bands 1 1 so on Friday ,comma maps colonel Philip IV and I like appropriate since the pianist infinity family tho I smoked the idea being that I do need to know about current income To make s twiddled credible but I don't actually want to change the then remarked the best way From and also an image of the organizations do isomorphic to that's that is wrong and Coke image Our isomorphic to the Colonel and call kernel of Diaz that allows me to dream so what is this as yet In my life and that of me I think that what he had to use the example of the examples right to the film is inherent operator where you may ask it's it's on the onto wings so the world Phil sections so if you look there I wrote down what action the section and local corn that's so this section went from epsilon III tool what was that I just wrote down the fiber last minus inverse off last of the and so that was my butt the section and In order to get the film section all that I'm doing is I'm writing down the miners with you the breakdown of talks with the Dubai my screwing up always lives in a age 3 are times once it comes from data very close to to the breaking it just lives over a fixed Hambletonian all the contact breaking the point being so that there's no on dependency and so when I take the linear the bar operated on there a general theory are invariant operators here if they useful if their from home at all than there actually is from office this is of the form DS and then you read my favorite capable Robinson spectral and I were successfully sidestepped the question What a Friend Matt between skills spaces which it's going have wait for like 1 minute and then you can ask me about details if you want so the problem functions that you need to to do you and the new directions and comes from a contraction and when you write down the contraction that you get from skills From the scales but what would actually buy a friend of section right moment usually we say well the mineralization spread that's something that makes sense the other thing we the duration works and that require a certain continuity off the liberalization and unfortunately this scales and so you get a contraction property but the contraction goes down in levels by 1 of the few started reading you you go further and further down to levels that does not bode well for any convergence so kind to the forces of back up so we L something called a contraction German property that sort of implements this contraction what is the contraction level preserving contraction In all but from many dimensions and so then he his of will hardly have approved lists all that so ,comma a From section is actually scales from means you always have a lot of levels but if you have a classical see 1 Paul but finally many dimensions you can do about implies this contraction property loss some small print and so that this can be found in friend paper and somehow I independently 1 but there should be a better definition of friendliness and and currently revising paper which I also attempt to prove that the property of trajectories and I would end by saying policy folds also because you actually referee ports so if the referees the audience thank you In the not complain of me to spread peace and harmony actually wrote the record and you from mistakes and pointed out things so I'm revising and others beautiful things With all the people who like to call for this is the Apple II I'm going to stand up and put them on the other side whereby examined all of this sort of essentially just skimmed from the user guide itself next year after year if you want this war than everything in what well I write now is asking me and I'm asking you because I don't know what the print it so I thought you know you can't function for 1 works by it satisfies whatever small print also as the index of the ministry operate yet and in ways it's building that's actually comes and you could ask of you define orientations saying it's constructed determined line bundle yeah so