1/2 Introduction to Polyfold Regularization
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Title 
1/2 Introduction to Polyfold Regularization

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Number of Parts 
36

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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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Abstract 
This lecture will discuss the overall ideas and challenges in regularizing moduli spaces, and introduce the two basic ideas behind polyfold theory: Making reparametrization actions "smooth" and making pregluing a "chart map". [related literature: Sections 2.1 and 3.3 of Polyfolds: A First and Second Look. Related videos: Lecture 8 and Lecture 20 from Wehrheim's special topics course.]

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and we to and I also think you and thanks for putting this on believed the world would be be a better place and I should also give you a renormalization instruction for any criticism that I might have to if I don't tell you that if there is complete bullshit I have not read it and I think it's not interesting if I don't hear it's complete bullshit just means you should actually understand the proof for yourself with and not trusted and by that criteria really think all troops in mathematics should be reported because what you should have sense so I'm actually not going to talk about analysis pretty much at all I'm going to talk to you about why we need to worry about analysis but also the triumphs of ground in the big picture of what we actually doing so I'm very happy that somehow the term I think that I coined regularization has taken root out but I wanted to see what that actually means From my point of view so that when you rise modular space wild and saying modular space and you're going to mean something more the curve modular space so if you can't find what you which will always be called and are you usually cake and like to science and some sort of unique way well 1 or 2 things so in the growth of which world I would like to actually do a fundamental class so but of course the question is what do you mean by fundamental flaws in the quality of something that might just be a compact metric spaces the right sort was a man of honor would have fundamental class if it is not a matter for we don't know what it's from plants it is but the nice thing with trying to actually construct right in here is that from the modulus because we have evaluation out and go to work and so if we hadn't Carson and then we could push forward directly with evaluation and that's why I think that's the cleanest way of saying what we want is to make a homology plant straighten so this is really going case and pretty much and all other theories as if the anything infinity related you're going to expect that have modular space boundary and so really what you want to do is you on single count things and then get a relation from the boundary and so what we like some kind of borders last no no not close but compact think and write so he you could get from maybe a compact manifold but at least that would have philology and Mulholland get into and bars of soap here but here I am right not because they closed because they don't have boundary contact madam folds and pride in general there may be ways ground and then from some choices I knew that Chris went up here had of him but then unfortunately somebody diligently removed this the task force next hour is also to figure out what's completely different in the philosophy that I projects here than it is on Prince's board about abstract perturbations out and I'm happy to discuss that in office
04:57
hours rights of these manifolds are going to come from some kind of choices of perturbations near the top of a lot more about the main thing is gonna boundaries in general and the nephew also components and the other bright identity usually come from identifying the boundary with some kind of financial fraud all from 1 the modular space itself so if this seems totally alien think about theory and what I'm thinking about heroes and packaging all Alpharetta trajectory spaces in this and Bologna so no matter between which critical points and no matter between of what next and then this just says that somehow the boundary is given by broken ones and back to USC what is this but that's that's that's why this is the philosophical views but I could do this up to a certain energy and then this would be true of all right so this is my goal this is
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what I mean by regularizing modular space in finding something that I can count on that I could integrate over and that has some kind and nice boundary structure the other bright and entities would like so what went on to write what might lose the him right from bright geometry than I described Jacob demand modular space is pretty of that have some varieties of some not manifold but knew they said subsets of an ambiance of bright variety that are cut out by some of the bright equation and so that I can do algebra to describe my modulus and the thing that ,comma realized was that while it might Meiji isn't integral I can't quite do algebra but maybe I can do exactly the same things that other to with PT methods sir that's why there's a lot of analysis you because all the magic you can do with algebra we now have to do With analysis however right we always have to somehow this intuition that Everything that's true of bright young should be true and that geometry and that's what it's all remove right OK Gasol said something's fullback yes this also good news icon I said Let's see what is my intuition he said what should I should say something right so 1st of all and an analytic and all and by now intended solutions of peace but then there are 2 other things and so I'm just going to have to think about having a fixed 3 months of this and a fixed almost complex structure on its impacted manifold but I have to do to more things to what about the parameterization because as an algebraic geometry and like to actually counted because it's not the maps it's so an image and by from in different ways by maps and so quotient of the ambiguity and then always going to want something compact so I'm growing comparatively occasion and so may be just as a side note to the examples that gave rise the beautiful examples of noncompetitive coming from Oct . running together however that's not the worst of problems right so I usually read the heart problems with Global Compact a vacation come from energy concentrating and that's where you get near the here bubbles but then you have an unstable domain and this is what really all analytical breaks loose and that sort of really appealing blow up situation so that you really do need this group of collective occasion even if we have no market .period sir if that's how we described and by their weakened while 2 things so 1st of all as you've already seen on American farms talks is more or less what local local from description together with glowing maps which will start talking more and more about answer very roughly what that does is it somehow the lie but the intuition that your writing your modular space as the 0 set of some kind of section actually mainly section here relied so but let's pretend rights To some extent that seemed to be the descriptions and reason you believe that the things should be the fundamentals I we need to really this find dimensional regularization there which said well let's pretend for the moment that my nice compact modular space of Jacobs actually cut out From a fight Frank vector bundle over a finite dimensional based that the I guess from making oriented vector bundle then 1st of all even without the section these things have annoyed at last so it's just the on the class of the bundle but you can pull back and that is said in the base and I guess ceremonial construct that with algebraic topology or you can constructed by taking a section and perturbing it and various methods of constructing on the analysis and so I I think the main 1 that we talk about and something that John the trees to do it by perturbation out so I would like to be able to just say that while classes actually may be the fundamental class of some section Of the 0 6 so that was very much said In the How right I believe prices actually all right so usually the base to be compact together order class at least but I believe you can do this as long as the compact all right here by
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otherwise it is not purely because of knowledge of the sure chance have that and also like I'd like to an an article on the front the men but With the final dimensionality is actually not the problem I think you could also take them infinite dimensional all and then you would just lead to sections of the press and then the same ceremonies gonna hold with exactly essentially the same proof right but what do I mean by this the rights of public construct what I mean by having perturbations that give me this island last summer and this is where I write is whether with generic comes and that's what I was going to strike and begin red from Chris's board because of the Soviet people always say take a generic perturbation generic Jake so if you give me a J Harman whether it's generic generic is not an adjective that goes with a single element of set somehow we say generic when there is a cold Meagher said and then the elements of that said of the generic elements and so on yeah so I think that's 1 danger because you people often say generic and situation was not even clear what the fun space should be from which you taking something and also said somehow 2nd category was wrong but I actually don't quite understand why I was so focused on having sort of a Cormier said a large sector full measure so I know what I'm going to say is really all you need is a set of not empty and a back sets of 2nd category are known and so in that sense that's a perfectly fine thing to prove and everything goes through it you just proves that something is 2nd category I just don't know how you have proves that something a 2nd category without proving that its core meager anyway so there is nonempty said in movie section so bad for any perturbation the sense that the good perturbations the printer section is not just transferred but I think there is still a compact yeah yeah we thank you L right so that tells me there it gets a smooth structure and that's compact the fundamental class and then I can convert that into the hands of the mortgagee Oregon a cycle there but now what if I take 2 different motivations right I would like them to actually have the the same fundamental classes and usually we proved by proving that the truth that's workable so right people often talk about the transfers allergy problem in a more and France volunteers religious respect where is somehow regularization for me when I say regularization meaning the whole package and in France facility preserving compactness and then having the uniqueness of optical borders according the right yet so I need that to be sort of canonical right I mean what I wanted to write usually action it's not so so usually the statement is not there exists the right the you need the statement that he would like to find out something L so it's a specific facts I don't need to know that that's cool meager I just need to know that that's it is not for example in crisis there might think that statement is going to be that it's a set of jails in this and right and the jails defined exactly by some friends but by L regularity right by its objective review of the the linear operator and 1 of I find it what conditions of his son is almost always going to be satisfied with comfortable with some of the things people if you like and you have to know he said that as a matter of time year out of the next sometimes you have to special units you know you look for in a how long this year you will also have a right to sometimes you'd like to know that candidates in east of office the output you could see you can write and you can also again you don't need this to be dance you just need to meet habit be dense narrative .period right so you could say take a stand and say I fixed the support and I think some of the size of the perturbation and no matter how small the Epsilon Delta are the same as always him so the yield on the back of a but images of the rights of the sentiment that given any J. there is arbitrarily close by Jay and that regular is also different from proving that something is called so that only 1 of them it was the right because I'm going to want to generalize this in such a way that it applies to more than 1 people in the world let's let's discuss that once were so deep down in their actual application I have actually factors the challenges of the becomes momentarily and then we can talk about whether different approaches will solve them what was going to say right so in this case you're already think they're so I think it starts Samantha and you would tell you that you know the set of ones that sort defined as set by the ones that are transferred to 0 I would indeed get a something however it's not clear to me that fall
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all the perturbations for which a transfer society I'm also going to preserve compact so in fact I think I can cook up a counterexample and then it's also not clear that you are always going to have the uniqueness of took borders so it's often you have to sort of and some extra things and in order to get the 2 other I and that's the magic of protruding J somehow but no matter how you put jail you always gonna have compactness and some between any 2 days there's always that a nice family from which you can run another golf modular space problem and and get the school board self we already see that somehow divisions of JI nicer than would be finite Frank bottles sir friends of the history of the the subject areas that people understand OK these local front on descriptions we have this pejorative description in fact we have a description of the order of all the bundle that's what a lot of the papers for a long time to explain and how hard can it be to patch this thing together this that's literally the how hard can it be that the 3 of as much of that was written in literature if there's any justice in the world and this directorgeneral wasn't such a way that it applies to homomorphic it I'm just saying I believe that the I really didn't think it would be that hard however there are a couple of challenges here so I want instead of a comment on the current she approach so if you really sort of forget about this sort of thing d bar operator you just remember that you have local Federal descriptions then you can write a modular phases kind of the union there was no drop any kind of ice across the EU the 2nd of 2 journalists and kind independent labs and so this whole lot of local travel operators by 0 sets will be some part of my modular space and once there from sections I can even take a final demand reduction that could be used to make these problems thinking all the time don't know what the final demand reduction programs should ask someone this afternoon means you can replace it with a friend you can replace the whole thing for the purpose of the 0 said by financier and ponder over finite dimensional space manifold L by brewing maps also fall into so finite ranks sections picture right so your local my description this kind of overlap right and you would think that you could make your class locally and somehow petulant together and how hard can it possibly be all you can think a little bit more globally and that's what both present to demanded is yeah action against 0 said of the Dubai are operated on some space of maps so here basis was maybe maps from signal and some bundled over the the section that cuts so most of my modular space except right "quotation mark the eyes of most of my face of my domain and then this is used compact so should somehow take a constructive occasions from growing topology so again I'm seeing this will set off a from section here and with his Saltzman there and are computer I can make this the manifold but as someone asked interested talk that does not mean that was group still lacks if I put in some kind of random differential equations perturbation here it's not clear that you that's going to be invariant and every parameterization it's also not clear that after I put this I still have a nice compactness result in the growth of topology sir you know you add some 1st operated to the who knows what might happen soon the qualified perturbing J is that no matter how you put to jail it's always going to be a "quotation mark variant of the sites and will present who was always going act on 0 said and for every job you have the goal of compacted so so really only works when you could object if you can get the transferability by pretending to some let's see the fact that I wanted to point important to say always is actually an extramarital affair section some old right so really what we would know in our dreams we can generalize that To get equity variant France's allergy wild for preserving compactness and that's just simply brought this sort of twodimensional ,comma examples to that all he had knows dreams we can somehow get no transfers perturbations in each of the little chance that fit together under the transition that but the transition absenteeism somehow continues master effective through this totally irregular space so it's not at all clear how you would iteratively construct that sir so we simply cannot you once you start this situation London AP you realize that it's really a little bit of a fundamentalist religion to believe that there should have a straightforward generalization to Jake general and while we can't expect a rate rise along the finite dimensional regularization by what why
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1 of the Act would variants right so "quotation mark so my compact modular space would expect to have a fundamental class Israeli fighter to 0 said what you a finite dimensional group and then compacted fight so if I now put by something To make the transfer fine then I have manifold it doesn't even act on it anymore right so this is not going there and you I think it will work in the a fall None of this has nothing to do with more support so seat I would want to buy so I need to write down a prescription for how to get from 2 fundamental classes and I would like to write down a prescription that gives me the expected thing the unperturbed 1 is by itself already manifold rights of the I'm going to 1 is isn't manifold then it is the and was observed what this group and the compassion fight is a matter for them taking its fundamental now if this is transfers and supposed to locate added perturbations but if I now simply forget the quotient I'm going up by by like 6 dimensions of getting even if this somehow magically was compact had a compact deification it would suddenly given a different dimension so clearly not the right answer right yeah right right so whatever description whatever definition of this regularization I write down all generalizes the transfer can that should be 1 of my actions for the fundamental flaws so if and bar is already a manifold it ought to actually be the fundamentals this is an important rights of that of you don't understand why we need equal variances squeeze people but yes yes right so this is there was a question yet please yes right that's what you do no matter how changed Jr the capital J it's like the ring from a transition was always connect on the other side of the company's exploratory wells we have been saying write well and I'm still differently right now not so frequently used the problem with this argument that the construction of settlements in the West action on the whole year gallery that we can talk more than ever many right so the main thing is and engage very gage who backed the acts and so he could do with some of the things I'm about to do and the parameterization group does not act in this way right were indeed to do what we can do about that while I am and I am and through here over I would the Fox I mean many this to energize you what wide with friend in these 2 weeks on I guess we've already talked quite a bit about the geometric approach right so the intuitive way doesn't go through we need to do something right and so that was the geometric approach that you c warning which has problems you know it works but it does all this magic and in particular gives accurate and transmissibility and because from that moreover there's something really nice somehow in and have automatic which is no matter what the JI to certify mine boundary of perceived boundary of modular space is the final product and that comes from some kind of bubble trio breaking compact fication then no matter how put J. it's always going to be like that this is 1 of the 1st because it means wall admits that really from finds some geometric way of proving the equations because these 2 things and I really don't you're going to need geometry in order to preserve the variance and compactness so there the various aspects of this right and
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maybe I should say and this is what actually but wanted who knew makes of more general perturbations of Jake but to me falls squarely on the geometric of if you right Hambletonian perturbations exam so prime Minister is it just doesn't always works right so requires some kind of geometric controls could make some kind of injectivity and by another side note here was the right people were 1st thinking about closed curves and the loss of injectivity means they actually dealing with of become and then you get worried about it so symmetries that Baltimore Crist called out of morphism group I'm going :colon isotropic guests which means you get into this or the full well and this is why people got worried about how do you actually make the articles of all the bundle however the ball when I'm gonna talk about isotropic this week and Angeline might cause at all because that's really not the problem ahead so Lack of injectivity is a problem and that right and if you still think that not somewhere objective always means multiple covered ask someone about the lantern examples of disks you have homomorphic no way to active but also not moderately covered and so it's really and for those transfers is really impossible to achieve pretty much so it's not so cemeteries and other problems that really need some kind of magic in order to get this going right and made rights of the paper to read for the 1st would say from all forms of and of course you yes you are right here and cosmic dust on the book of the Bible against big but also right and I think it's cute but warned it was not gonna write things where I believe that if you have questions about it I can answer and right that means and it's not right now on the board right now so I would like to answer any questions about for quite a while all I believe I can answer most questions about the others tho and this is also the most readable of all these accounts and because it was the most readable Hassan was the only people which make mistakes were actually found and that's why this is the 1 people that didn't get published however I believe that actually that the 1 that's closest to being flexible or at least I understood more often than not anyways so so there are essentially 2 versions of the the crime and all of them on the ground approved and he to do something that I would call we have I think called obstruction bundled me for what I really think much stabilization approach which both using the of congress they aren't you all they are easy but don't
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apply to so there's a whole bunch of papers about abstract ways that if you have a section of the this kind of structure
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then it has a class and I think those are all correct except I don't think any Jr modulus these objectives ever gets cut out by section like that so that's what this piece of criticism applies however the stabilization you really think is the right idea and this is very close 2 more conceptual good definition of an oil accounts rather than by pretending so the Polly fault approach I believe the board should have discovered Polly folds 20 years ago if you don't think so carefully at the point where mistakes happen however the Polyphonic drops of as of now talks a lot about perturbing I believe 1 could make the best approach rigorous by sort of using quality folds and stabilizing and then there's this best perturbations in the picture ,comma so what happens here we still are going to use this right idea except were actually going to do it and thought every time so the idea is to really get ourselves rigorously into the setting where we have just 1 section of 1 1 the right thing that was that was the idea that I really would have liked to describe my modular space but as 1 section and then I would have liked to climb my regularization the question is what can a bundle so and what about been invited so that's where language is coming these things were going to need to define the key is that these things have global structure where is the stabilization approach of the but only had locals instructions and what else right just like before I should assume that I was told so if you have a topological space you could give it a locals and structures but just putting shot local woman from foreign and if you don't assume that the transition smooth veneer of locals in structure but they don't match and that's exactly what happens and in all the papers if they always like this there are small structures locally in which we operators smooth and its rental they just don't matter right where is probably a good number of compact metric spaces that you can't even find local remote start sir once we have set up as just 1 section of 1 of the reasons for his run for office there and hopefully we will not not true but right now statements in a similar way of regularization the amount they're so pretty much were discriminated defined thing here in such a way that a they applied to both the curves and be regularization well that's the goal set in the regularization there some I'm going to write songs said of perturbations and roughly speaking this will be contract perturbation supported near there was some measure and for that nonentities perturbations again to properties for all I have some notion transmissibility and compartment and then all have who was right what should groups right to now and it's cheap so I am going to only talk about the the political version of and when this is the Empoli for version we used to hear the fly but 0 said is in fact a finite dimensional manifold my boundary and corners but that now I can say what it means for the 2 0 to pretty much exactly the same Sara but except now generalized to some notion of bombing which hopefully applies to the all the obvious pro is that there seems to to be a very clear way of actually good regularized modernize spaces and the obvious con is that there words here that going to need to know so you need to some language over head but that's why so yes you the political L yes that's the right side of the problem here a was very good gram of Whitman Varian and the negative trend last then do explain why multiple covers will no matter what Jay you have will always give you things that are kind of transfers sir also if you if you had don't 1 Warri about at the moment you don't know whether you curves were injected really have to rights yes I write I call that someone the right so you rent right so the moment bubbling so soon as the bobbling you can't actually use domaindependent almost complex structures anymore because because the body is not part of your original demands on the bubble somehow automatically J is not domaindependent and so if you can't exclude say sphere bubbling you give forced to deal with the question of somewhere injected you had a hit you may be right but then but then you can only do that if you don't have boundaries and is not clear that the baby don't think they can do as if to do so right now witnesses totally "quotation mark points on this photographs of its and you will have complete references by the end of this week so I have the other ,comma but it is so that the boundary of the it is still far from that of someone like spaces of the port of that is going to require some handson construction so "quotation mark right here France and pretty much right this kind of says that many the perturbation along the boundary of mine Polly full should be
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so that the project off the perturbation but the thing is when it is allowed to them constructs a perturbation like the usual functions that it doesn't mean that on some strong arguments given by the product of some father that's that's something you need to just do handson and that means you come to define these perturbations of iteratively complexity by complexity Apple sometimes are possible and sometimes it actually is simply not possible and then you need to sweat a little more interesting things happen so that's the master approach right and so evidently right now I need to explain to you how this fits in in with what we think is actually the modulus but we know how to describe the modular space as a 0 wants something and then come back fight we don't know how to describe it as the ascent of something so so the talk of paradigm shifts happening in not From describing the modular space this is just 1 0 7 what happens here is 1st please the cost and then some of and breaking the law sir he where realize they can't get prevent France's oddities of the perturbations opponents to happened after the quotient and it actually also needs to happen after adding problems and breaking the audit and Polly folders that we 1st quotient then add bubbles and breaking and eventually would put sir that means there's something weird going on right on this side have a bundle my devour provided the whole thing is accrued and under my some of his group and so on this side when should sort of do was I should divide hope that gives me a bungalow I hope that somehow my section still makes sense but not just that I can and need to actually sort of add the broken things before write my section and now right don't take overlying should maybe make it a weekly line so this is not a closure on any topology because those are never going to be compact and they're going to be very infinite dimensional a function space but I need to add all the stuff that I eventually won in my compact but what United States beef or I saw the equation the of people yes we do "quotation mark In my interpretation of what you do you do some hope well let me let me that this is what the yield on the issue of the journal I thought that you don't know now no no no I no heard but never mind so what happens here right and this is the core ideas of political theory my my theory has always been that if I explained to some granted some reasonably smart graduate student these 2 ideas that I can lock you up in the dark basement for a year and you're going to come up with political theory so but let's see so what we need right so far what we have move structure plot a Space Command what you know the isomorphic let's surface despite the fact the actions is pretty much never differential and that's what you'll hear from you this afternoon why this is never differential so composition with maps is really really bad itself so this is pretty much the only way ways in which this is differential is what you do your eyes and prison group is actually find which doesn't hold its signature there is for example the S & S true or it's hard times as long to thunder yeah but Toros all of these will not have finally Trubey groups I group and the other way is the map it doesn't actually move in the final demand environment for manifold purposes in this match there is also nothing different from otherwise I challenge you to find any reasonable trichet space apology on an infinite dimensional space of maps and which this in 1 point differential so why don't you exactly yes this that this is exactly what algebraic geometry is easy was 1 of the reasons because you don't need to "quotation mark but because instead of you there was no immediate Kulka the analytic that's right yes yes all right analytic methods might be L by the end of the year will be defined the idea is to pretty much defined notion of scales smoothness and proved that this action is scales right will they scale the structure on the state so in that sense I would say we pushing out of the 1 see I know what you need to talk to them about that we talked about it is going to the architect of the boy to have you don't you 2 minutes to go so far this is a reasonable idea to have the 2nd idea is really not reasonable that so I ought to do 5 2 to describe the name of a broken or normal as if it was sort of and abundance
1:01:08
manifold somehow but I
1:01:11
want to I'm going to revert to told example to use the right requires some maps from time to time to find man from this 2 plants the data I hear from my final manifold but somehow nearby I'm going to have it really just 1 man from 2 to and a different on manifold from that In particular the brand name is described by something "quotation mark and that should be show the 24 of the underlying political all and political b sir which fortunately also gets me out of the problem of fix of matching notation was Dole's could not immediately invitation sir what happens here you go somehow prescribed nearby could by previewing maps I'm going to take writes described all the curve so that the business no point in my big and nearby areas concur that are still total but I've put both maps but there's also no because nearby so I described them I actually rightness of rotation parameters from them introduce and then there's going to be some parameters which could be infinity and that's not let me forget about this 1 here for a 2nd so the new boy now and another nearby mountains and going private and I'm going to make a freeze nearby which I get essentially from writing the water with a long neck and 2 with a long neck and I'm I'm going to forget the 2 beyond and anyone beyond here and then interpolated on this little strips and the 2 groups together sir that's evidently not going to be geomorphic anymore but this is actually the answer to what is the topology so this tells you what neighborhood the you rights in this case there should be a rotation parameter in this which has just ended election great so the Kia observation is that this map is supposed to be a chart but it is clearly not inject I couldn't for getting data idea answer the idea is to say well in factors through something a really weird stuff said here the but then this effective something we haven't Oklahoma and this year is something like a retraction it's a subset in here and controlled controlled way of getting from a general point here into this and that's really the key to happiness is because while this is a we face it has an ambient apology for this ambient apologies to traceable and it's 1 has assumed structure and everything is good unlike his microphone .period In fact the mission would be the this year after right because what you sketch sort of said I could hear but but then I have to preserve Koreans and compact according to you we have to describe what it is correct yes but which means I need to figure out what my ambiance spaces and I want an ambient space for my money the face of the compact modular space actually and that's actually that's sort of a general problem that's exactly the problem the problem of ambient space is what we end up having to do with current issues structures would be also end up having to deal with with the I can and see that approach is always what is the right ambient space in which I could describe things as just 1 0 you know what you expect me to do and organization the of which functions like a process for such mail but in the end users could give them away if you do onedimensional manifolds yes but but even a onedimensional manifold in order to get something from a friend section onedimensional you need an implicit function there but you need a notion of France's allergies so as you can probably get away without smoothness he considered maybe see 1 just for the implicit function there but the beleaguered technical now so sorts mail is the reason I believe you really need smoothness and not seeking I want to be obviously I reverted to all the people in it and I think it he explained to a student might might claim was that if you give somebody the definition of scales movements and the definition of what these retracts off like the now all my and graduate students of course but it was or what did you want the only thing why did I but I didn't I don't want to do is trying to explain the rotation of the group also writes technically if I should write this 1 times this because at infinity no as 1 barometer so what I do at definitive here as I just don't and when I don't know I don't have an excellent and that in fact is the reason why this is not a boundary .period it's completely an interior point the going to see I think tomorrow jobless cannot explain to you that really full no longer really the employing parameters just the disk and by the middle point is Infiniti SUV so here pretty much if you will think of these as Basques right then you don't have an 1 parameter and then you see where the boundary comes from but when there isn't as 1 parameter than infinity should come without and all will will have any anyone with like philosophical questions of like Why can't you just I'm most happy to talk to you all week and no particular may well have a session this evening when we think about why can't we just I want to hear be