Introduction to Regularization Problems

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Introduction to Regularization Problems
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Moduli spaces of pseudoholomorphic curves arise as the zero set of a Fredholm section of a suitable bundle, and one expects and hopes that they can be regularized in order to define invariants that are stable under perturbations. This lecture provides an overview of some of the analytic difficulties that must be solved in order to construct such a regularization, and briefly explains some traditional approaches to their solution, namely via geometric regularizations and finite dimensional reductions.
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as In and you have to do good too while hello everybody and I'm going to give what I hope is a very elementary talks so you know where you're probably very bored but for those of you who don't know this or have forgotten a little bit and I'm just going to go over the problem of regularization and explain their traditional methods for dealing with this a little bit so upon the question Is this where we have a syntactic manifold and and it's got almost complex structured J which is only time and it was looking at giant home Nice to be simple I'm just going to think of the demanding experience flying right say and told her hot 0 came well decorated by all these things clear in any sale which is going to be the set of maps and I'm just going to take this serious-minded good Maine In Richard J. Hall movement which represents on the class paying so much that's a clause in the 2nd homology of em and and Laurie I'm going to have some not . 0 1 through the head fortunately because underlines the and these points .period different and that's just the space of just of maps in these
elements of maps and the detailed recently the elements comments plan and as everybody knows what all these words mean the government assuming you do and if so if you don't please ask compared to you around and then we won't we interested in this space but also were interested In and if you got to elements using within sight while it is equivalent to another element new privacy trying to enter the diagram of this kind so we've got the From as estimated as to which is the isn't undermined group so that's a lot of bottle of matters but you going there too bad because you Prime going there and the diagrams communications there gutsy trying them and that goes to and feces but so this would be me something of the former and you try will be the composed with and this is the future of the epitome of this equivalence relation I'm not interested really In the space of actual maps were more interested in the space of equivalent of maps so that's going to be redness and talk 0 so that's going to be and filled 0 apparently divided on London's equivalent to the the equivalence classes so we're not really interested so much in the way the parameterize just that they have a parameterization were interested in images of geometric images and and I put talk here because this is this is a top strata some completed space now so for
example well so case but this end till the talk repair of and today it is a snooze month-old found transversely content Oh I forgot to mention it was jailed Norfolk while I forgot to say what that means is that there she Green and operators is 0 so that so don't want and you know it's a hot and steamy homes giant into contention so that's a cushy mean operators and so this is 0 7 operator saying transversely contrite meaning that the that the derivative the operators suggested by going to say more about that later on but anyway you wanted to be a transversely cannot and on the dimension of this thing that would be equal to the index of the operators to end constitution Donovan and plus 2 which is the number of Mt .period considering a two-dimensional space that's that's a dimension and why you wanted to be transversely contract if it is a front on theory and that means that if you very J. this place is not because so that's the that's basic space of general moving maps and on send best case the 1st situation is that this is a nice manifold and then as I say what were really interested in is the question space so this is on top that this this is almost never compact that is compact enough unlike the space is almost never compact you can see that it's almost never 1 thing I'm insisting on got .period the 1 Susie Case and then went to the old different from each other so children can come together and out would that would not be in the state and then another problem about that compactness is that this group of the columns this group here has the see is not a compact group so that means you question the out and buy something which is not compact and that can cause problems but what do I mean by compact enough well sort of food for the purposes compact so complete was 1 of the partners as well for interested in the evaluation from Antilla well relief from and topped 11 world and tends to put in and unimaginative and to the caring so we've got a pair using which is an equivalent calls on comfort brackets like going to the new evaluated to see 1 up to Bloomberg remains intact and that of course is the final displays maps but it it goes it could not sue court so well-defined map here and what you want for this is that you want the image image is so that it regretted to represent homology calls I understand that you see in and this were compact manifold and of course it would fundamental group classy would just look at the push forward of the fundamental cause but in fact French represent homology logical it doesn't have to be have a compact image when it needs it is that the if you look at the closure of the bridge so this is at the root of the evaluation not minus the image itself so this is the boundary of the image they should have "quotation mark dimension at least 2 so that means what what we say is it represents a pseudoscience and To give you an example let's take very simplest example is I think really simple example take projective playing with the standard J and classes to be the possible long and then but then see what the question closely aligned with me and so the dimensions of and we take to mark .period but dimension here it's going to be it's too added cost to see 1 and a half to carry which is foreign and minus 6 which is the dimension of the repairman transition course hinted at OK so this is convention you a Celtic see movies group and this
is 6 and this is for and so we get and that of course is the same as the dimensions
all protects the consumer times Cpt so if we
look at the evaluation From the end talk 0 2 in the cost of allowing going to sleeping 2 times 2 early that way however newsy goes using 1 missing too and what this is releasing represent along and so this is that that the elements here are representative of the next 2 years got 2 points on it which in fact we can normalize can take the first one to be it 0 and the 2nd 1 to be at infinity said because the group is strictly transitive so we can fix these 2 points and then we still actions see acting on these things but this is this is and the top wins with effects .period Center 0 infinity and and pushing out .period still it's still something of space rock squirted out by noncontact and that any we've got this space not Europe looking at where they go invoked his points and in seeking to a new isn't the lines through room was a unique alliance ready to points therefore this map is injected please take standard complex structure we know that injective but of course is not subject and you can see it's not subjective because the 2 points it's going through of a difference they have to be different because I'm a scene but these 2 points to different melodic and got so the image is equal to seeking to clients seeking to minus the reclusive not compact and you can see the space not complied by on hand the set of maps is somehow compact that's why go lost here and you can competitive autism is diagnosed convention to it's it about cut real convention for exports because that's where these 2 points into diagonal here all and and so it's it does represent a system that has a father does represent a fundamental class because the image of that that that the boundaries of small justification "quotation mark connections delusions and included in the inside the In in terms of the case so we're looking at is this summer not talking about the structure of the main here which is to a good cases you can assume it's metaphor but it's not going to be compact but unambiguous saying that incident it said "quotation mark Vermont metaphor but if you look looking images here it represents has a fundamental
cost because it's because the boundaries of small loses the please Bloomberg News said there was no mention of closure but was a culmination is a real ,comma mention of at least 2 in and to the Council of dimension that have so in this case and is a New China in this case and that what's what's wrong with this it is hard said this is 1 her ruling coalition allies the icy wastes yeah OK so what I mean by the US I'm saying this incorrectly what I mean is that that the that divided the image minus the image should we the Union of maps From submanifolds the and the Soviets and this is some units and the dimension of the should be but less than or equal to the dimension of our original space and hot -minus 2 that's what I mean I'm sorry your yeah rock that makes us so it's a column mentioned means it should have at least 2 dimensions than the dimension of the and he was involved in the deal and that's assuming and top has the expected too much which is going to have a fit so that was that was my 1st assumption that it is it's a transversely contact mindful of the right to mention site means that that that means it has has a different dimension and so on and then the right that you're assuming that that this thing is going to represent this is meant to represent the homology class all the right to mention the conventions that that's that's OK but so now let's say so there several things here and the regularity problem is the problem of arranging the 1st thing that it's transversely comes to find if you could find a jury to make an announcement that the regularity problem but then there's also the compact probably wanted make sure that this image is compact enough for how can you make sure about that and that neither compactness is actually much easier to deal with the regularity so let me just say a few words about compact so was that this cross compactness theorem which tells you that if you but that if you have as you know you look at the the the the the sequences of of Sears right and you look at this this is his inner city from so we have these so again soon we've got some kind of new cases from say as to and and got a sequence of commands and their ended in effect similar to and so that means that it is at the that the pullback of American although it has to do is fix go unpaid Digital and this is actually analytically it's expanded of its own equal to forget the fooling around with the pullback of DU square and the energy you put out basically the L 2 normal view it was measured with respect to the correct and you have to choose a metric on the basis of measure it the correct thing to get that this is the energy so so this is a sequence of calves so these are the infinity maps hold because they have bonded I want to know what it is that actually W 1 2 this is a L 2 normal the 1st derivative so this is what's called the oven it's a sort of case if we had because these our home figures the whole of it mounts and all the two-dimensional status if were violated W 1 keen on 4 key bigger 2 then the standards of a compact arguments that tell you there would be a compact subsequent "quotation mark convergence Center that's just standards elected soon that's I'm lying in the W 1 Pinot my piece because into his because this or if you have something behind it and only 1 funeral that controls the C 0 so that means it's electric continuous family indices urinal so you sort of standard results the continuity to extracted conducted subsequent but what we have it's borderline cases where 2 to not not because so in fact it's possible for this thing not to have a convergent subsequent but if it doesn't have a convergence of sequence what's happening is that the end of June is somehow concentrating .period and you can we scale and get a convergence of the of so that that comes comes down to the fact that for example in this case here were cordially I've been here buying on compact and on noncompetitive and so that sort of affects a converted but anyway this is really well understood what Gromov fruit was that there existed some sequence and yet you also called UK said that convergence To win well-chronicled McCusker what we now call stable enough and want a stable matters ECU bubbles can develop and that all explainable within this kind of thing wrong but what it's going to be a is the domain of signal is another little space and unknowable spheres it's going to be a union of CIA joined the points they could look like this could be a bubbles use secret Service feels like that that will be the main the base amount .period on here where because your mark points would converge somewhere where you got 3 more points they convert somewhere like that and not you with their nap each component into absolute have a sequence of billion however you have a occurred a set of maps and that's what it is so for example just suppose we were looking at a map from as to intercede which had agreed to so that would be a quadrant but it couldn't quadrant can degenerate into 2 so that's enclosed to that's 1 generation could happen you can have that's a geometric degeneration where you have a sequence of things representing a quarter at which degenerated to but you could also however the generation that happens because for example suppose you in this case is not points come together will not .period among like to come together it here if they do come together then you could have a such a situation like this where you actually have a bubble and that bottle contains both Mt .period and here this is what's called those component it is uncertain that new restricted to his component is constant so what you have here is a situation where you have this this is a long period of amount of aligning your 2 more points sitting on the domain it in this business sphere which which is called the ghost component so I should give you a proper definition of a stable not so you can see you can put it all together into the spaces stable not only is there any way of retrieving outboard I wouldn't want that
to new on the other they think Scotland very told person this this is wrong In the name of the up this thing doesn't that wasn't what this is this was OK the deal think the world and that's that's good enough look it's going up by itself I'm very obedient during the latter half of the year when I was 1 of the injured the material you all the well and then she just comes from the from the map so revered doesn't see that it doesn't see whether mop . 4 but you you so I wanted your definition of the stable man and course these can of hard in any genus there have to be due to 0 to restrict commentary due 0 for simplicity but as only just due 0 because of the point but Jenas 0 is isn't you can think of your domain as a union of these 2 and each you Yukon from tries in nice when standard complex structures you don't have to worry about variations of the complex structure on the tepidly you'd have a noble suffers which 1 you have to a lot of that the complex structure on the surface to various types but you can put that in a few months but that's just make it simple so it's a it's a it's a noble notable women surface so you have something and it's all it's meant In a reworking of something like that but it's meant to be connected in a tree .period to Malaga have something like this that would have genus of religion is just 1 of sorts of it's a treat to treat label trees and you have a certain set of Mt .period sprinkle your points and Bryant and got five-year pointedly sprinkled around like that the month .period all and so their own particular spheres so you have to say which summarize many lesser maps you which is the 1 that each component and so we have it's so that's it so we're looking at 2 polls which is performs Signora the complex structures understood here is the view and then you put in a call while the audience equivalence relation that just like you have before you just have to I have some hold will think this is a whole Norfolk map here so you're interested in these modulo reprivatisations I am and then as important stability condition I don't 1 you don't need to when you talk about a stable not new doesn't have to be home office I mean that you were interested in the whole of over 1000 generally just look at arbitary maps with with knowledge names like this and but it's very important to have stability condition what that is telling you this is equivalent to saying that for each element if you have signaled you and you look at their 3rd the group of automorphism so it's not so that to itself this is finally the another in don't like yourself to have an infinite number Baltimore and what that means is now that this say on a component like this component the market non-constant then the matters by either injected some multiple covering and the could only be a finite number of emotions that I can take you have a component whether Marcos non-constant there's no condition but if you have a component of the matter you will constantly here then you see what what would an automorphism exists we will have to fix nor was the 1 he would have to fix this noble .period but see if running got to what it called special point of only got 2 special points on this component this is not going to miss the nodal point and so there in principle is the 1 complex parameter family will system so that would not be allowed so the claim is that if you Alpha but restricted to the Elvis component is constant S L 2 Alpha has at least 3 special .period and special points are nodal points or more .period that's important so that means for example in this particular case I could allow that to be that could be a ghost components so that 1 could be a ghost component this 1 could be imposed this was logical that undergo to that we couldn't so this would be a possible degeneration if you were looking at 11 Maps I'm from Nero from asked to to see continued the so that the kids in the
class twice line then this 1 could represent a line that 1 could represent line and we've got 5 more points now we have really good stable In bit of this company and claim is is that if you know it is solidifying and plow 1 0 10 generated to date the equivalence classes of stable maps In genus 0 so this is a genus and came up .period and representing the class and if you got a jail the their budget because that's the space of General Norfolk stable Martinique currently it's a stratified media Wybro top isn't that talk was its top component if you like where where it's the main just Henderson was Smith's fierceness here but then it has all these other components where the remains of a noble cause and some of them if you like a sort of artificial like this is an artificial 1 because the tumult .period come together and so in a nearby nearby things to this would be something where you had and new hence this would still be constant saying this 1 could have easy to like but now we'd have seen marked points on that compose so and when these 2 markets come together would have this bubbles of artificially created in the domain this is missus conservatives idea way of describing what's happening where describing compared to attention the plan is that we have this the space here this is compact it's a compact sport and if everything is regular so in other words if all the underlying maps that you see the regular saying would for this would be that this man was a of not transversely Colorado 203 more about later so that and then all these intersections here got spaces the interior spaces using abuses ,comma concept all these intersections in the Transvaal so that it'll sort of comes out and it's all in the right dimension and the it is the structure the formal dimensions From the strata is equal to the dimensions all the original of the top space -minus twice the number of people the Ancient knowledge gives you contribute sort of accord mentioned too I and that's a formal dimension index so that means that if you can arrange everything to be transparent I think the transverse that will be the dimension to but across it doesn't have to be lot but if it were works then you see if you're in a situation like this where you can range of all the lower strata have basically have the right to mention then the talks saying this will represent a serious like because this is compact so you couldn't have survived the image of the man by just looking at the images these things and then the boundary of it's going to be covered by the images of the of the higher dimension that strategy the height could mention struck hero and they're all going to have at least mention too so that's a sort of standard regularization picture that you competitive on the image by putting in the images of the evaluation maps from the from the strata and this and this and in this space stable maps if you can arrange if you can control a dimension reasonably is then and then they all have conventional used to and then you get incidents like now of course the kicker is you can't always control dimension that's the whole problem but if you could then maybe be then you beat you solve this problem you have got because our aim of course it is to try and use spaces of maps to he gets me gets variants we want to get these homology classes out that we 1 intersecting at golf with endurance and that this is just the close curves but if you're doing open purview you have the boundaries and you put boundary conditions on the formal structure is exactly the same equation looks a little complicated because you have to deal with what happened at the near the knew the non-contact parts of the domain but formerly it's exactly this idea that you can always compact to fight the space looking at by making some space stable not so stable buildings or something and then you just need to control the dimensions in principle all about components added should have coordination at least 2 you need to control and make sure it does so dense tally was going to say about compacted and reduce the time most of the race here III erased the 1st condition which was that everything was transversely that's what I need to talk about it because that's that's the hard compactness is sort of the year playing his land while the unstable match would be well at this thing is that they don't appear in the competitive exclusion because if you have an unstable component stores that were over an unstable components so that not was constant then you can just forget about that mobile and put up with them you see the unstable components of things which don't actually contribute to the image because the the map is 0 it's just a constant menace and even just collapsed earlier the unstable components and if use it's certainly not a good question arises because he is the I'm saying it's suppose we have is a situation that this this with it was a constant map then that would be an unstable components but it but the thing is that you have is so I'm saying that that's not going to be in the compact vacation because we don't allow that and the compact conditions what we what we do is we it not particular
case we just get rid of that completing in just put them up .period there wouldn't be a bubble talk because the image will be signed during the sequels I'm a lot like you justify him so that it was stating that they were the stability you when you when you look at the compactness argument what's happening when when above all forms so that typically I mean it this in this particular case that this is what we had with all that would have happened here is that some of the points would come together and that you know the underlying map here you just have a sequence of maps which was basically converging to a map and all that would happened to create such bubbles is more points will come together in a particular way sort of these 2 points coming together faster than these 2 points coming together 2nd amendment would form the spoke and you can actually see that completely clearly just by so the rescheduling of blowing up and looking at a nearby this you'd have security the point where GE scaling this point is going to go into these bubbles you'd have a busy forums 5 very very close and then sort of further away you'd have as the charity's strain so the the 2 scales of public here this is the 1st these want to be the father of further ones and these ones are being later won but the map itself would be constant that mattered and would be constantly talked about how the mob .period go and it turns out and if you true this exhaustively in our own broken jet whom overcome fatigue if you just look at the way improves the compact you can get a you could describe the army of course you could put in something like that and a lot of as this 1 as limit but then you wouldn't have a sort of unique and it doesn't give you a good structure and business was conservatives realization that the good structures because it would be an infinite automorphism group which you wouldn't want you wouldn't want you'd have to divide by the stock here you out there set of automorphisms of all these things to be final referring to review all of leaders of the other people in the world still has a cease-fire although annulled because you see here at the Madison on Const so you've got some bad news and you're saying you equal to new composed with the few had also Sofia would have to be something that fix these 2 points and so you'd have to have clean because he wanted to be 1 the of the nodal point that's a infinitives infinity and it would have to satisfy them well if you is non-constant you wouldn't have it any way you could call this the "quotation mark will eventually you only have a finite number using what you could have you could be you could be a not so you could have new from you could think of seed units in addition to an antique take the exercise EQ and then you will have an automorphism which took accede to the to and it too high with he is that would be an also was also the you of the wooden but it would be a finite group cannot say there aren't any old more than just a it's a finite group of autumn so if it was a multitude covered her it could have a lot but it's an interactive some are injective so that's another thing you have this idea of somewhere rejected you 1st have to and I don't know what that means is that there is some point Signor such that if you look at its image and then it's community that just single so that it's a matter capable this here's a domain as to his use the on this little neighborhood it's injected and then you see if it's interactive there and you have something that behaves like this that you have the key would have to be the identity in that little neighborhood where it was and that doesn't that doesn't happen if you have a mobius transformation Texas 3 points at the idea so I'm really this stability condition whether so the geometric Graham understanding it is telling you that the group of automorphisms of this stable not final that's what the stability conditions and that's a that's a very important condition 2 eminent physicist a fact that you could have a lot of all residents means that you're actually In other space and this space here the structure a typically older faults rather than Monfils because you could have all the more so you can model on August 4 but it is very important that the final those United's overflowed with finite undertook the history the hearings in less than 5 years of guarantees civilian action all OK well I was saying they were from class and all right so then then Omega varies a number right so they do have financed and you be alright I mean if you don't more general sayings the fire likeness of the energy is actually what makes this whole compact work you got have abandoned the energy it that bad on the W 1 2 To get started with he is the only way to do what you want and watch something like that hello could have something called the runoff topology which I don't particularly want to go into hiding its each stratum it's clear enough I apologize and then nearby will is this structure you see when you have a when you have missed as stable met with with and I have to mention this at some point but there is little the element beginning anyway if you want to know what neighborhood of something like that you have to do so after you have to say gluing is aware if you've got a noble domain as owners of nearby tourist things which nearly have the bubbles nearly have the knowledge that don't quite fit in a nearby using here we've got to know words so we've got you know 1 billion from the here and now that we've that and for each for each interview have you get a slightly bigger board so got a single note so here is a we've got some map so
this we've got let's have no .period as simplicity this is going to line and that's good sometimes so that the nontrivial not so this is going to paralyze said the image of the apparent Will nearby this paralyzing simply to do actually have of course quota impossible to draw
but that how analytically juicy that well you know you have this sphere you take you take care of this fear and you take with minor discontent take that sphere minus a distinct thought to belong to this group was silent you got another 1 as 2 -minus suggests the size of silence and then you shouldn't believe that binds together you identify the boundaries so that you have something like that so I think know we dismiss it out of its sky at nite when you do know about is you have a twist here you have featured twits so that the parameters you have here you have an upside on which measures how big the the disk is here you have a seat so that you can think of as many as kept silent went into the 2 features that is the it isn't see that's a little complex gluing parameter so when it's Ireland 0 is when you have the bubble and when it sounds a little bit bigger than 0 you we've joined and remains together now Of course in practice what you cylindrical coordinates here so it sort of looks like an infinite density would you think of something looking like this and something looking like this so you take away a point here and you have this infinite cylindrical and and then you chop off your at some point and joining together and then because the map city How did you get a whole moving up here while the matter you've got new 1 hearing you to say there so well at the near this point it's essentially constant so you can just on this new together thing here you just make your pneumatic good you told us that at which is you want you to game together that will go with some viewing any it's basically equal to you want here is equal to you to hear and then he just patched together with some kind of function in so this is almost home of not quite come off and then use Newton process to make a formal so that's analytics saying concluding and again at the moment is very worrisome case of Mountain Home cookbook not doing you know you can the analysis of this is pretty well understood the it's horrible I mean it's real analysis right if you're colleges like me and you don't like analysis and loving and otherwise it's a sort of pain that have been so long but it doesn't exist it is well understood and anybody can be done in different contexts and goes in the front home In the end the political contacts have a very beautiful way of doing good but L that and then that's what gives you the topology money there is something called a gram of topology that you could measure which I really don't want to go into but basically you can take if you have a man and that's universal with the names of books like that you can you can a basically well I understand that you have to reschedule restricted bits of the domain of the skeletons of measures I don't think you want to go into the says is a topology which come into have the structure and each node so if you have a strong team was 1 would like consisting moments like that then having over that is a line bundle consisting the
doing parameters and then there's map which takes you that long model into this results so you get even get into college good grammar and developed topology don't need to you could describe it without the good about anyway I don't want to celebrate about the regularity questions of his assassination right here national this was just the original series of exactly yeah so that that's exactly that that's exactly right with this process and you it was a very honored by should say something about regularity and justify the price of the OK so OK so and we II had the notion about some were rejected ,comma touchy right up here now what is transversely cultural wealth so we have a lot of the commissioning of Doboj but is a half of the year history ,comma Virginia combined with Algeria and last well the you is a differential so that that we're thinking was the 1 this is this is a wonderful on this on the demanding With coefficients of pullback and attended Monday's visit and the Home Office 1 the deal itself is a wonderful the fact that we add that to make it a little and future home office and you can do with it and you can with just and and so we can do that so that we have we we can think in general we have excess some space and maps perhaps I should write W 1 peanuts from S tune into them we've got as a slice of taking 1 was complex structure and we have a parabolic similarly have you so we have but an and you doesn't it and so that this is just a matter of minutes and check and then we have this operator now this operator let in it it's announced 1 former see its image depends on you so that means that we really have a bundle of here and with the J. as a section of this fall and the finder of this fund you just so that for each paying and here we get a map in there that's precisely the status and I actually if I'm completely missed W 1 peanuts and you can do this in the sea infinity case if you want to do in front of cereals and other put some ballast spaces here so that this had better be the inside of the indecency infinity things let's have the a mail piece sections of the differential treatment peace but so we have this we have this bundle you can even prove this is a nice bundled over the space it's about when we got this operator what we'd like to say transversely out is we like what we're interested in 0 7 and what we want is for this it we want we want it we want and dulled by to the transverse to the to the 0 6 so we don't them at institution and so to say transverse that means we have we have this operation we can look at the mineralization L of nuclear which is the linear rises in the mineralization of this operator ministers met for about a wonderful development fold you differentiated C 1 should be the 1 so you could differentiated and then you could project onto the new project onto the fiber and using a connection in general but only 0 so it's well-defined and then the lender isolation is sort of the mineralization also so free evaluate that on the other side of wives side is the tangent space To the mapping facing you and why is in their hands and faces betrayed the space of this complex structures this in your eyes operated takes the form of it's served on the liberalization of the Of the crystalline operator we've fixture just linear Isaac and then you add to it and that this sort of linear time coming from varying Jabal J. actually peers here only new citizens half of composed U composed projects you have to have the liberalization of assets of these 2 types now you fixture that of course is time goes out and the transversely contact means and believes that the Community Church is subject for all In the solutions and that the monitor using operating here if you figure look continue fixture and look at that operated this is a afraid home operators so that means that has finite dimensional Connell has a close range in the fundamental Coke so it to say it just say is it's objective and it just has a Connell the Connell has affixed to mention equal to the index of the operator of and then it's a it's a very it's an open condition that it too had to have subjectively reservations of dozen implicit function here in this spring's context for Fred home operators so this objective for all you hear it and a very gentle but it'll still remain remains intact so that some kind of nice stable situation so that's what you're looking for and its of its objectives that means a solution spaces mindful the like mentioned is using implicit function but now height you want the basic theorem assessing the basic serum the setting is that you can't unfortunately truth and we would like to say that if we allow variations in jail we like to say that it's always a member of but that's not quite true so the ceremony but if you look at em please they include the services at all some were injected j hold because you I I'm Argentines understated and firing OK well that that would be helpful thank you and so you look at all summer objective :colon and you align yourself to the injured 2 very generic this is governed by a man I suppose I should have actually I shouldn't be even little more careful J. L it would consist of Cl almost complex structures for the better ourselves in a situation of a that money for rights to a better like the Infiniti J we better completed to be they could be W 1 .period couldn't you can or W people you can make them to see almost from the start so that's about it so that's about manifold and then if you lock there Evans no just taking a pair you Jerry the but suggested sorry that this is fragile for
eligible studio here it man she hasn't always fragile but really what I want to say is if fell is large enough agenda that doesn't that there's a set sitting inside the of regular values and so that would be where the singer is transversely contact it's right it's called regular forcing his trends as out and the point here is that this has it is 1 a residual reason you will understand that Boris follows you can say :colon meager or a furor in the 1st edition of the book we did not we sent a 2nd category that's actually a wrong signal mean it doesn't mean what we meant to but anyway this is a large set in other words it's a accountable under section of dense open so it's a large sector of the economy good thing it's not open dance itself it is accountable and 2nd opened and the media always empty lots of stuff in it and in fact is trick of tides that allows you mean principally due this when you have a CDL saying that you can take L equal to infinity here avoided by centripetal tells so well that's telling on and then so that means that there's a lot of 2nd hero of regular J where the the weather cools solutions of regular and so that about but the kicker here is the parents have to be somewhere inject you can't do this you see it's a multi become occurred you may not bill to make it renders the difficulty is that if you have but have you from best to have and this could be a regular that 1 could be faster due to be and it could be some but then we could take a map which is a composite of we could take the form as to itself out of the car and then look at the composer reviews so perhaps I should call this sort it's not a whole so this is not this is a multiple cover Goodyear during periods of this for example could be you that's into the care know that if you have a regular column that exists and you could have generates critics still exist and persist so you may be able to have you coming on the altar this may be a part good regular ,comma the challenges you always have you then you always have the small called for what's wrong with these multiple colors the of the factors of the In all this well it's too and passed to see 1 notification plus 20 well cuts which we don't have any money .period tenacity to but now the difficulty here is that if see 1 a prison negative which is certainly could be a Munoz no reason that in order to have the index index of you used to have class tickets to see 1 of them and because if we're just looking at map from from experience and it always if it exists there's always a 6 prominent family because you have older than me promise translations so the index of this has to be bigger than required to say In order for the for this comfortably there that's because your intervention in order to end his tax you could certainly have this being tenderness and minus-4 and you have you have a common cause with the negative tone but then you see if you look at something like that this thing be so negative that the actual index could be next and yet discovered always be the unity competently to worry because the underlying purpose always there and therefore multiple columns so if you have if you have a structure in the stable markets which actually contains something that's I mean you could have a occurred class but have coming costs would being and then could decompose into something of course the miners Katie plus case where C 1 of the reasons that they could be composed well many you see you're necessarily bills get rid of the constant close community then may always exists there could be multiple columns of comes for lodgings but then the index of less than would be big because you see the 1st John class here Of the C-minus Katie will come 1st classes he has negative effects of the 1st Class a B minus case bigger than the 1st of so that means you'd have a structure in the car and B-minus Katie would it In the best situation even if you could make the more generic they would live in coffin space which has got to that means you can you lose control of the boundary and therefore you can't necessarily Yukon Yukon have simple methods of getting rid so that's a that's a problem why you can't use regular standard the thing mine why do you need this to be indictable I don't have time if people don't know the organ like explained to you can ask questions about I don't have time to explain why it has to be some work to do here but basically the reason that you got his own manifold and destroyed the very major so when you varying using your your to their here to get regularity but your very on them by the image of your views as to the way where the domain of your matches your Mac is going to some wonderful honest so you have to take a variation in jet which lives in and 28 . 2 points to not that summer objective than it's going to be a little neighborhood here you can control it control Jane here this summer injected you can pull it back in control of but it's not summer objective when you pull something here it would have settled for you here and so you can't it doesn't give you so much control over what the variations objective something as I explain it it's not summer injected the kind even really have the bad components of the compact the courage and is too bad that can't be dealt with now I was going to say more about the analytic difficulties in dealing with us In the in the traditional way you see what you have to do I'm explaining some geometric regularization you after more complicated variations if we need you know you have to allow yourself more complicated perturbation which don't just depend on me conurbations underlying was himself here just to very J which is stays of the big variation just on the mound for advantage on the mound for when you can have much finer variations of just To correct but then there are all kinds of problems with local and federal appeal the office hours in the evening was the 1 question they ask that the people anywhere in the world but that is what the that and in yes it does the 1 parameter families because everything so they haven't had mentioned too well well you don't actually have to know it's connected to the regular what you need to know where is that there is a regular family warmer regular home top and regular home of his seat you have 1 from that family here that's what gives you an extra dimension because you've got a variation of Jerry one-dimensional families so you can have a 1 from the family of things here which doesn't actually consist of everywhere regular things but consists of things where the code of the code dimension of the images just 1 dimension right so yeah that part of the Syrian tells you to regular values you can find a regular home atop in between where the inverse image would be and then the recommendations the city has been going to regular now you can't necessarily pass between 2 regular just because could it would mean that the space John Wolf accounts for J 0 and forgery warm word I some the basically the prime and basic you always contains a couple this spheres in certain situations you can't but In many interim general Yukon the question is what they 1st


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