1/3 scBanach spaces and the sccalculus
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1/3 scBanach spaces and the sccalculus

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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 
Polyfold theory is built on two new fundamental analysis concepts, and this talk is focused of the first: the sccalculus. We discuss how in general the action of a finite dimensional smooth reparametrization group on typical Banach spaces of maps is not smooth (in fact, not even differentiable), and then introduce scBanach spaces and the notion of scdifferentiability. Two key results of this talk are that the action of reparametrization is scsmooth, and for scdifferentiable functions the chain rule holds, so that many constructions in classical differential geometry functorially extend to scdifferentiable geometry. [Related literature: Sections 2.2 and 4.2 of Polyfolds: A First and Second Look.]

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2 and world and during most of the fight a preliminaries bid to my talk here of ideas the 1st thing to do is just apologize to everyone that we actually have to to follicle much is on the same day as sort of feel as a little bit cruel but you know what can you do the 2nd I want to point out is that so at this Web address here it is true that this outlined all the colorful talks and it has that information in particular that abstracts that can be found by just websites but also at the summit lecture notes molecules for today's talk for instance are on there already and tomorrow stock and Sunnis from what might talk on on Wednesday will be there as well hoping to add up 1 writes about catching lecture notes In the summary readable fashion icon that those up there and I'm fairly certain allowed home it's all of their own but by the end of the week so follow along in particular on the last I heard helmets talk was going to be a slide talk and so I should be action will have a slice of their as well so anything goes too fast you can quickly backtracked on your laptop right had to see what he was talking about so there's very stuff on there in particular in the homework problems those already posted bond and references to the slings for instance all they still the papers on powerful so you can sort of quickly access of variety of stuff that's that's referenced this approach to it right and so that is also briefly before actually get into the stuff talking about you know before I start talking about this a calculus I think you know they're these words being thrown around like you know powerful colorful theory so I think it's a natural question asked kind of what is a policyholder have some very idea before we started continuous so I thought I would I thought I'd write this down just as contend the guiding cases were confused and so on so I answer this question was employer following answered vaguely becoming diesel Indians would do that as well as I fall it's it's like a bonnet manifold on museums that has charts and so forth that has potential boundary and quarters and it's a suitable ambient space to do this regularization procedure is kind of on cyberspace of like space of all possible maps with the right topology on it which allows about you know no borders and allows on Kurds in Rome of winter developed knows that allows for trajectories to break and in particular it's possible to build bundles over the use for which the the because operators a section and reduce regularization procedure so that the key ideas this was like a bionic manifold with Beveren corners but it has been a structural steel well better in some sense more general structure which allows us to do a lot more things and then and also this key point here is that I am powerful is to a manifold with powerful is to or before right so users also have was that it was between a manifold oral Falwell essentially is having this all local group actions and in the same way that's the difference between an envoy told Polish also this the discussions this week are all going to be about involves and it's built into sort of a larger more general powerful framework next week when element so too did in some sort of idea as we as move forward right so and that in my lecture here I want I I I wanna talk about SC calculus and so to get start with a motivating problems With myself as to when it was finished products here's a motivating problem when it considered that the maps that continues differential maps from the circle into the realign a realvalued maps from this 1 the group action and to find his group action but essentially saying that well g applied to state S ,comma is equal to as plus whatever the amount so this essentially is to shift map in the in the in the domain of your function f In saw Oregon because this is as reprivatization action as of now I can say Well claims I think I think probably tell you this is the most important part of my talk is that gene is not classically this means as a as a function and I classic not pressure differential for instance so that we can make a strong statement which claim while homework exercise of universal cruel so we can the special especially don't think you know what analysis for this might seem just a touch overwhelming so I want to give you some ideas as to why it's not anywhere differential so you should start with a bit of internationally rights tuition should basically be the following if I look at this and say OK we'll want to vote very Nunavut fight at during but if I then I unvarying in the domain of ash In particular differential with respect at a different with the then I end up having to differentiate my argument and that's a bit of a problem than in particular if Athens just the 1 and no better than I try differentiate right there and try different you respect as they have to differentiate efforts and to minimize space and have problems so easily into which right but we can be more precise than that because of the stronger claim or different so why not so let me give me ideas so I while step 1 is to say let's suppose it is different then the during djp what has to exist then it can be Media computing directional derivatives so that ridiculous to make that computation is Suellen DG at that point as in the direction while at the request of attention you can see that this quandary here has to be equal signaled that prime the great
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OK so so what well different ability and guarantees that the jury with respect to does the director because different so the following the following equation must be Sigma ,comma fight since 2 0 0 in 1 topology of the plan 4 it is essential for to have a director of pieces approximating so this this inequality here has to be true however we then do it was right by some sale the following 4 so high here it was 1 of the point here wants to this point and in particular as a consequence see the slow clear wonderfully function flies some signal crime is equal to mood minus signal In here by promising is equal to save and then all we want is what the these functions this family functions to be in hour in C 1 so it's a round out the corners just a little but that's point as well as the relevant the key factor which is the fight Sigma Phi sub signal was 2 0 with a 0 0 in C 1 no words about what this that this into into the right hand side here in the past and I don't 0 1 this is not just on the assumption that when intended and that's actually not too difficult to check by Soviet border thrust to the top you can go and try and work out this example and then the point and the point is that the finish off by my claim the G is nowhere dense has started G is nowhere differential is just a matter of of shifting the pointer I like this point margin so not too difficult especially since he was a true should be the case to the work necessary to prove origin and include just emphasize as I suggested that we think is the most important part of my talk is that this this will prompt action here in this toy this action is not classed difference period so you if have a reverberation group active in the space of functions In general that that will not be in the classical sense so something has to be done Hi catch investment but support has been a long time slot Starbucks work against making for something so so I have a question that piracy stuff for the 1st time I might some questions so I am told problems but said that we actually class is also relevant to anything so the following the following examples here .period In his appointee you when you can put isn't some something manifold are and what everyone right hands the pontificate introductory this is in my time limit to pay part time limit to Q and particularly find some some space of functions it's consider the space of a C 1 functions mapping from all are In 2 quota my target is Arabs named W. such that the action .period the gamma here such mines and people's pain and damage to the right 1 online I want to do that is to say for instance you have something that some some more function and appears more standard of 1 critical . accused and 2 critical 41 categories trajectory between In some years once and for all and space running around the course in that while this problem that arises that that is well if I have some idea which is going to be solution to the greater flow of problems then I can In the domain by 2 end other solution and the like and I have always family were of of repression of repression of a single solution which showing up here but the cost of doing more knowledgeable you want the county's things what quotient up and how much quotient and if you justice well naively what what might I do however not doing things were too complicated and would you do naively I think once considered doing well to patch I use local patch are you in the I choose a correlation 1 manifold locally anyway but knew image of the image of the trajectory would for which this trajectory crosses through transversely you know I can do it so well that I can really do is identify these to be able to get in such that the 0 at present age the cuts
16:38
down my domain a lot right this this allows us to essentially chose assuming at this point in here is is is injected and was not even make appropriate modifications of this cut down function space by exactly 1 dimension particularly looks like the least .period is that this is really be but model or where I'm thinking of all positive under me which cases far so great and the good instead turned out in fact these debate has an eye manifold structure to it you can put a bundle of the town can talk about the sections from sections that bundling shows the great of the great equations and gives you a friend Home section perturbation by the family stuff at this point right then you say OK well now the problem is that I have essentially the same about metaphor which depends on my choice of age so would you want to do is just sort of doublecheck really quick but you know it's the case that I chose a different shades of my hikers services in general there's no there's no natural economical choice fiber surface here and if that's the case was limited to a different flight and landing is going to make sure you know these aged prime is welldefined so you know that's
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fine that's not a sure factors that of being a nightspot manifold to write and say Well this nightspot manifold and this 1 of menopause was probably check to make sure the transition matters and so you so well I write down a transition maps Alaska is something like of the age into beats prime and it necessarily has to have a form of the following types 5 of the game is equal to gamma composed T sub gamma plus whatever my point is that if I have 1 function of 1 function in in summation maps 0 get smacked into this hyper service I would keep the same image and ship that along for a reprint rights would only the now Gamal applied to 0 is indeed prize so is my transition that's no problem you want to see who school the problem the problem is that well if I differential with respect to gamma this season but is a real privatization action which depends upon him so different it was put to get that I have to differentiate function here with respect to get the changes that have to differentiate the entire thing with respect to go on and on and on and in order to differentiate fly I have to differentiate my arguments the problem is exactly the problem we had with the 1st example that I usually that that was not classed the difference during service in area where I have this function I have function defined between function spaces and that case that I have to in order to differential that function I have to differentiate the arguments that function that can be a problem for class French and so that shows up here and it shows up it should have been the 1st problem ineffectually it shows up in all my July problems this this is a problem that needs to be resolved in any questions about that so far have here what was about to be structure on right so that the point is not to the point of running around here is that at the end of the day you had this regularization this regularization idea that Katrin sort of the get the Kachin outline of In order do that to say Well you know but what I really need to come out of that I need some sort of implicit function during the show so whatever space and make use of water and space spaces bundles are you going to have I need to have a function at my disposal and in order that the the most now I think the most NL analytically easy set out to do that it is 1 which you dealing with bionic spaces spot manifolds spot bundles and so forth that's the easiest way there's hard was a function tends the Yukon do any kind of they they share a number of similarities with things that should easier done but my understanding is he really wanted was a function and in the right way to have your dysfunction terms coming from you know considering Bachmann for its press In the light of that at that time so the question so I suppose is also the issue of of orders so so yeah I think to reiterate its national Crist said I think that if you consider your space of if you consider essential to speak your mind MySpace whatever might be grateful and in this example order was all Kurds you 1st considered that and then consider the automorphism group acting on that 1 like that in this action for sure but the difficulty is that whatever you do 1 of those spaces are kind of transversely what we need to have in order not compatriot might not be compatible with goal of no no curves for instance the might back because good grades broken trajectories for instance and know what a larger share Indian space in which to 2 in which you can sort of perturbed and have this essential realization that capital suggest right so the fact is that because that's the case that we want to build a world large space or make operations and that's why that's essentially what I'm focused on on the presentation of the problem .period classification questions already you need to know about this this is 1 of the right to work so the
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2nd year raising further questions no I mean was saving you end up with all the same problems that no 1 is going down during the I'm not going to be fine I'm not going to be here this is not the only 1 of the the cost down the leaders of the downside is that in order to do the powerful approach you you do have to carry a lot of sort of analytical overhead with you new ways you can see the need to know about as a calculus and retract since language fits together conceptually I think it is sort of easier but end the Invalides standing easier than the various Korea she approaches on but there is sort of additional overhead you kind of have to keep track of to keep track of his ego because of good OK so so there's a problem so what's the solution is on the idea the is 2 the difference so how to do that so well what we need is this definition of an 4 you 0 England into goes you can walk into 1 because contained in this way can quite well in into the 2nd 3rd and 4th on the inclusion In 4 the the Infiniti is defined to be equal to the intersection for K in the natural numbers of the the proper UK before trying these properties here they are you trying to do is fairly simple and say Well simply maps from last 1 2 are 2 realvalued maps from the circle that regularly see maps so fast signal
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into to and on March the
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matter so much say didn't too 2 for all the years of
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this a close friend sir things you'd want from OK so right OK so good and so how the 1st step is to replace of space which is was completely your space with particular collection of complete space which satisfies some properties relationships once said that the idea so just leave it at that for the so now so now I would say what I said in the the idea was to change of different abilities so now we have to define what it means to have continuous maps between expect that the maps in a a couple of ways to save the his skin so as to shorten 1st scale because you had scaled by taxpayers in this terminology was used by I can put some members of the was that it was from serving the literature in her elsewhere interpolation thank you but as or if you want France's scale whatever scale but space acquittal doing this sort of conventions have completely been established yet but that is or as ceases 0 provided L continuously for all clear and the natural numbers I would like to think just the sort of claim might think this is just 1 of space and what we want to think about the perhaps this is a constant so now we have a definition of continuity but we intend to win get the definition of depression building just a 2nd but I have to mention a few properties 1st 1 property that if you have 1 of space Annuity directs space structures and might expected property you want scale spaces for constructive on the scale spaces but essentially by forgetting a finite number of turned off the front end of the day and so in particular identifying superscript 1 of the greatest and levels of unexplained superscript and can some number of of of all levels so I can write this you want the case level of the new 1 space assuming he uses the bonnet space is equal to each 1 plus case and more generally even then OK level of the yen is equal to you know what I can expect from this properties to have and so on this is more definition and property of a listed here anyway to we new contained in a knot is an open society the structure by the time so I can point out that this issue here is slightly better than would mean there was a need to have in the structure on 1 sort of a topological space which is linear space and I complained about this a lot to prominence for authors but they did they never seem to clarify this point an incident in the user's guide at continent enrollment and all over and I wrote I would try to clarify this a little bit by bringing this notion of skilled topology is really interesting that you know an attempt to make this less vague sort of makes the concept less clear so I think the easiest thing to do is just allow this ambiguity inserts a well kind of what's going on here is that having opened sat in this a base level of of other scale space are also ends up having this filtration and it has a lot of features that you wanted sort of local properties anyway so are local certain local properties have been induced on these opens at its best assistant leader will be vague the property as you based on for this has to do with the requirement which over here and 1 international space OK any question so far so the
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tangent bundle of you some skill 1 expects and unfortunately it's now you is actually think it should be In particular megawatt all do let me do the following is a social I want a tangent bundle of of of an open substance containing some skill 1 space this is a strange thing right I write to you is equal to you want cold blustery 1 shifting scale structure and the following but then I think natural question as the station this you know why it is equal to actually you plus the former natural light and so I this is going on because now is the time to bundle is defined everywhere it's only to find those points which hot something or have at least 1 of regular so words if I if I jump back to my examples which have now been erased all are always some examples of skilled Bionics faces we said for instance but this set of CK realvalued seeking maps from the circle this nite it's OK suppose the base basis with the continuous maps with the tangent bundle to this well it's only the tip of the tail the 10 bundled basis of attention but was only defined at those points which earlier see 1 not C 0 straw and I can't really tell you why this is the right definition of take until I tell you about but what it means to be scaled the French and then this definition become more clear that's a good thing I think if you're confused because you're released following a long enough to be confused yes that was the right note that's another point you'd also think that this would actually be 1 up here but it's not In so if you want to write down if I wanted to write a letter nuclear right if and was sort of by space or what the structure of the has to be you want yeah I can hear what but the question would cut the question is what are you mean the approximation asserted to watch easily approximation any right this sort of implies I think the notion of a notion of different ability and we haven't talked about skill the French building that's why said was sort of a necessary component of the tell you why this definition makes sense right in back reminds you how we know that you know I mean what's coming up to Olsen 2nd here is the reprivatization action needs to be as the smoothing petunias believed as the 1 and there that sort of situation where I need to build them differentiate this function in order to differentiate functional differentiation argument between that lose regularity and that's that's essentially the key that the secondoldest restated a guest on but I think the key ideas to why you need something why you need something like 1 of the look at what time they want 1 of you have to know that this is the 1st of the standard of I have to say that he had you but there were no and all I don't know of any other examples I it yet and a kind of it also makes sense I think is of no interest from home here because From sections in particular those arising from from different repeatability pity on you know they behave nicely in between on and work nicely from say CK plus 1 to seek a on the side that sets a bad example Sobel class solo regularity squandered social regularly of K and images from arbitrary can map itself doesn't change the fact that works nicely across all these levels also sought an indicator of Y would make sense to have to track of all the information they behave nicely scale the blood and is exactly what I've written I I don't know what I mean yeah is in some sense out of I would argue that what's happening here is because had this sort sequence of Bonn expects the sequence of bonnet spaces you really have a sequence of topology right and and on different topological spaces which is dense and
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it is in it every other sort of space is below and so and so and so that's kind the way I see it this is just a sequence of steps all arising from having a sequence of of open all arising from from taking the intersection and they have their own topology of the apologies for Iraq nicely in the same way that topologies and in a the of 1 of the it's about the best I can the director of the lake tell me what right maybe it will but you have to have a local social services abilities and are fine space something needs because that shows up in the definition of the definition of the differential you can just look at proper Brock has to have sort of a local infrastructure take .period inside the build small point of of small actors and state spell I want lots of things that I like about it but again but I think a point where it really makes more sense to say that makes more sense to tinker with roots and what's really know how houses is being used and that allow for the ambiguity because might try again to try to make precise thing necessary becomes less clear what want so so it's particularly by tended to think that you would actually entire bionic space in this restriction myself to a neighborhood inside a large about what so now I think I can provide a definition here which so the cost
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differential or received 1 provided 1 4 you have 1 the linear operator which alright the which depends upon your domain .period X opinions and their operators from you 0 2 at 0 I'm engaged 1 minus 1 of the Act next traction 0 normally divided by 1 of the 2 0 in the 1st condition In its condition In the past 2 years which is defined map from the new continues to the given by why last year the number of that spelling words at the University of 1 this year the year of the new year a mail it's all about the contents of you know so the questions yes but I will is so here and there just the 2nd 1 1 bit of of of 1 piece of information here that I think helpful as it is the 1st time I saw the 1st time I her several times but you know it just seems like this definition seems kind of technical and looking at the head of the the Justice and the UN easily overwhelmed by by too much information was just too complicated to me right so that the 1st before moving on the 1st thing I would say when asked or perhaps remind what around 1 . 4 a month To be differential well you need 2 things 1 you know when you need it but and approximating come and to it is something precise this is the use ideas in the classical sense what made clear that the same thing I'm doing the 1st statement is guaranteeing the existence of a linear there has to be an approximation to function as there is free enterprise has yield approximation function the center with the limited said so this is the 1st step is to guarantee the existence of approximating linear map and the 2nd condition is essentially demanding that those things very continuously in some sense the new notion of continually cut continuity showing up making use of the scale scale continuity and serves the classical continuity which requires something different right maybe if you believe this is sort of the classical sense initially spilled the stair these 2 definitions and say OK well I don't really understand all the analyst everything that's going on at least I can see condition was approximately a map condition to is not sustainable linear maps vary continuously in some sense so now so I can do I can I can try to destroy the question which is why on the doing and so I think it it's instructive to consider a couple of examples the 1st example example but useful consent fight came to be called the cake was 1 of the ones that circle that case is equal to ck and then in the past gets map too the prime directive as the 1 now the cheaper this kind of a cheap example of because it's classically differential but that's raises a good point which is essentially that of Europe if you have if you have a function which switches which is still continuing and is classically differential at every level that is going to be as C 1 so sentence essentially classic different ability morally classical dirigible is applied scaled Frenchman so we are generalizing the notion of a different different abilities but again in terms of examples as kind of a cheap 1 must do something more interesting the of caves war 4 the fact that you have to prove talk only 1 possible really actually the privatization yellow bonnets space to everything is compatible with the definition of value can change as 1 of the workers sort skilled manifolds beloved altered but is often a problem what approach In a letter notes
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that I have on my advice he's just to get started issuing moving in the right direction this sort of work it's not really important because I like I know I think that this is not it's some not in the room of buying see but I think I have I have to we know and many of the right so why this example important so this then goes this goes back into to the very beginnings of motivating example that we had the motivating example was all we have is that function space and we had this reprivatization acting on it and the key factor is that the most important thing that that's the thing which catchall tell you the most important in my in my talk is that is that this is not classically differential simply is not vastly different on that causes a problem and cause a problem because it shows up anytime you want writedown essentially waiting time wanna give up something like a manifold structure To say for instance that the Obama space of maps
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model repressurization act this shows at a time tried on transition that's the sort of action by repressurization shows up and not have what we need something else and now this sort of tells you everything .period that now we promptly action is actually at the different it's another sort of a whole pride of course the idea that the the things that make this up as the 1 for instance also sought to tell you that the corresponding maps of setting up the Moss analogy Flora marjoram or whatnot consider those types of reprivatization actions or also as the 1 in fact we can prove that the SEC smooth right so I can say just give a quick definition is that is that after mapping from 1 of the 1 as the bonnet space to another is S K if he 1 so the derivative is sort of no 1 what has wondered degree regulators so I have to tell you something South associate something kind of important I guess which is on so the important thing I guess is that so what we do with OK look it shows up a lot of his remarks problems that have this action by repressurization not classic is sort of out of a scale as they should think a hot you know we've solved the problem by just changing this notion of different ability announced that working with class the differential we now have the scale differential motion as it seems like everything should be great but you can't celebrate yet the real concern is if I would have a function that I like to think is differential but in fact it's not and and and then this person tells me although I change the definition of a different abilities now everything's fine you should say that the Seattle himself so what you really need is is a hero the fire it's essentially the channel so the general says that you pass this amount from Egypt energy is a map from having are lifestyle G I think I and think this is S C 1 and this is also the 1 that in G ' S C 1 and not the other day accord the corresponding not tantamount of decomposed is actually TG exactly what you would expect and now now we you have to change your disposal essentially the essentially anything from say finite dimensional differential give different and that anything the finite Mitchell differently on training construction you can make their that doesn't involve imposed function there and then carries over into the scale said so you can talk about scale manifolds skills smooth manifolds you can talk about scale differential functions between them you can talk about you Brunskill differential forms there's there's quite a bit but you can do and in particular because in the case of arrested by now that 1 of the key features of finite dimensional scalable spaces was the scale was always constant it turns out that anytime you're dealing with finite dimensional manifolds the scale calculus of the classical so we've really done is generalized on this notion of scale calculus from this classical setting from of stories about manifolds yet yet yet we generalize this notion of of the class different geometry and Bart manifolds into the scale but manifolds with the chain rule and so this action by reprivatization is not as he spoke unless there is good so so so so conferences that there's sort of 2 key features that knows more graduate student if they were given this sort of thing is locked in a basement for a year to food and water in the provided as well but no 1 could produced through the whole theory and have the story is half the story is the 1st step is that even before you allow for breaking anything we had this problem the problem is that fight the problem is that if I tried or run right down this quotient box full of maps from or into a manifold between 2 critical points function and tried "quotation mark up by doesn't have a bomb manifold structure before you talk about breaking ball anything else doesn't have about manifold structure and the answer is no then have scaled manifold structure as an example want to do before right or what before was cut out by services and services were cut by 2 different labor services on transition absence Walden fast but not the scales and so you can things look promising and what the next concerned with how the breaking ball with glowing analysis of that decision tomorrow so right so catchy would tell you the sort of 2 key ingredients sort of hard to to to the the whole powerful machinery to really knew I wanted to have and I would have would the effect of theory actually constitutes a half after rethink what from theory gives you in in a sort of classical setting you have to have it provides a different definition of something that defines the same thing for you in in the classical setting use and modified definition which is sort of quality standard was poured out over into the of the Polish scale count framework gets it does it does get more complicated but there is a version so as an organizer and a reminder speaker time but I do want to mention just a couple of things which is it what I want to try definition if you already have and tell people worry about the future of definitions that only got knows how many approved something is scaled different was as the 1 little scale is really hard to say something I don't ever want to write itself To some extent that should be true for most people and and and and sold right but not by much more than that that's essentially the essential point I Look here's my function is moved it right is seen in the usual sense we do hurry to right well you develop some basic maps right say Well you know I have some you know I have polynomials for instance have got some products compositions these sorts of things and approval of the what these cases but what you already have chain rule and disposal and you already have the thing which classically is also essentially scales right and then what happens when you're talking about proving things for more than wife structure and some of the more you have to go actually used really make use of the Internet functions were safe or analytic they had to do was difficult to what you have to do this is painful painful but it shows like kind of functions smooth something which is constant for your on that increases up to sum up something so this is a hassle when this but once you have this now have at times of other things you can do right because the interplay between things approximate you have a lot to lose your balls and so I would say that if you look at it still busy at the smooth newspaper this would provide to all the building provide you many many building blocks essentially I would say all the essential building blocks that you need to prove the transition maps a R R essentially as smooth and inappropriate they give the building blocks and those difficult things like the analogy of the high cost function is classed as that the they have these sorts of arguments on sort of for these sorts of of of all of the building block type functions or and these building buildingblock functions are the ones that always show up in promoting safer right and what the complicated a lot a lot of transition for instance which should be the a very complicated or smooth was breaking apart into bits and pieces you see see that all the bits and pieces have been shown to be swallowed up by Italy's he already so making use this stuff is actually as difficult as it might seem and I thought I was going to provide a list because I'm almost 10 minutes over I'll simply say that those that list of sort of properties and references in the literature are in my lecture notes which are online and on the Web site that stated that the very beginning of class and that only but I Trapattoni started something of a based on the shift function shows up on the question 1 of the things that you want them to be the people of the United States this but I would say the point is that if you look at his estimate estimate has a hole in the 0 level so it's sort of like saying right better about a linear map announcer defining it on you know once incidents said for instance that the balance have to be .period 0 all along on the London Eye the numerator and taken the owner of the I'm trying to find out if they are to you this is the you know what you do is you 1 of the reasons for doing so In the use course of the year and the the question of who owns the you're we don't want you to that know that we will could go on and on and on and on and on and on and on and on and on and and on and on and on and on and on and on and on and his on and on his car a lot how do you how do you prove that the way you prove that I would say is by taking difficult cross slices and then joint transition to smooth the yeah experience in the United States I think that you know I don't have answer to them but In general think general I think there's a calculus really hasn't been fully exploited I mean I think right now you know everything there is to talk about throwing the 1st week here is really what is you know what is the analytic foundations initially using needed in order to do everything for SFT sees this is this is sort of what we need here and based on the way pieces for 250 element sequences OK well 1 really wanted to do you know for Kai category stuff for you know or you want to do relative as if Teri onestop 41 sort of attach flow lines to things you know that there's a lot of things which actually extended fit into this framework believing that I think doesn't fully doesn't fully exploit what what issues you've done here and I think there are a lot of theories may be outside support could geometry where this sort of stuff can actually be applicable but you know that's not being looked at so strongly at the moment because as if to 1st and then so all recommend recommend that any further questions be asked later this evening or tomorrow mm