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3/4 Polyfolds and the construction of Symplectic Field Theory
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Title  3/4 Polyfolds and the construction of Symplectic Field Theory 
Title of Series  2015 Summer School on Moduli Problems in Symplectic Geometry 
Number of Parts  36 
Author 
Hofer, Helmut

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/16296 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  Topics: 1 Strong bundle structure and the CRsection as Fredholm functor. 2 Polyfold packaging of the SFT problem. 3 Smooth Multisection functors and smooth weighted subcategories. 4 Construction of sc^+ multisection functors. 5 Auxiliary norms and compactness control. 
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00:00
there was a time that the I think the religion itself but it's so so so far so throughout the category of stable romance using the phone but there are certain things then some of the the subcategory of automorphic objects were moderate objects is given by the we just Associates rate 1 to such an object another 1 0 and the idea as to the form c . J intimacy .period so that's more of that properties hold and so we discussed the 1st mature than the 2nd lecture we showed actually that that is the slowest structure on this category and will explain what that means and what they would like to find out moussa switched and now how do we get them out and that this going to start to happen in this election Is that militants of define another category lying all over for rich the cost before but you have to question operator gives a section front so the finalization of spaces for what each of the 2 little experience and the courtroom commuters a section of this and then the seats are very obtained in such a way for the nozzle from other kind of form dosage of the standard find your which Ukraine viewers monitors sections another explain all this and this the episode here we already know order of some kind of small was objects so soon was formed also reveals of Gunderson was formed Europe and if you choose this saying in general position satisfying some properties like this but we have to formalize and for listings than this 1 will actually have this property and will of a good time namely it of the suppose vacated category talk so locally itself represented by many forwards divided it's not so so it's good enough that you will but the thing that you can actually integrate forms words and so on and actually can define SFT but there are some issues concerning users such something that was once the would have to discuss many on television and on television all the more bitter than should go to a covering the reduction use numbering just punches and so on but so that is what you need so far what cases is minor
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categories so we have a category of stable maps and take a fundamentalist bases of so that took so that men former formerly the things that I usually into somewhat general formulation about 4 but for the category of stable matzo you can think of all the countries supporting this key works actually fall the most serious so so we associate with object is about space and in this category the morphisms so to different objects might get 2 different spaces but the morphisms are actually lifted tool many isomorphisms not so then you can define a new category nearly Texas object and the vector of which flies over the object Nunavut spares and what our morphisms loads appear 495 is the more physical category maps these belongs To the available space lying the Zeus also the source of fire is form and and what is the target of this small fears it's just the vector obtained by applying the to was the for the victory carefully so you left so you lift each morphism as a linear the fibers and the object of the victims in this far and the target is the image on others like 1 so we have to go but not of the that and so on but we're going to so concerned and want to destroy the use of violence it's likely that I will be the budget so you have to undergo a of a prime said you 0 0 0 there's a more physical fight between the the and Zeliff off I deserved the isomorphism mission would madness here 2 you and this thing here so we can identify the morphism for this bond category this the underlying object and the vector the said what is this the the source of this is the and the target is the image that you get so so now what do we do in our market is well if you are a subset the building of wide 1 that saved and what do we do better than this then he the McLarens class of maps up to I action so we take a representative now receive takes attendance space hero then because you have the option you can just identified these are crossed the tangent space of the underlying the company and of the snappier Utrecht as they are confident and the and the the and and all of whom work space for free for those under consists of all nets which are complex onto new off from the underlying agreements of the Ponzi into this thing which is identified citizens are taken representative of your model action with the 1st factors independently defined on which they are end of and this map should have a certain regularity property namely it should be away from the lords of the class of each tool there and that is because accosted on sexual act on its Hristov many ghost Ondrej and at the ponchos shows you have to also take to take exponential decay of this thing so if you didn't take a puncture undertake cylindrical from all 4 corners to want explanation that the decay of the derivatives of Boston derivatives of or 2 that is precisely the new take this stable maps Johnson told cylinders and gold there have quality Sweden 2 0 loss was returned sponsored portion of the evidence is exponential decay and then it applies equestrian operator that would go to twitter artisans the syndicate so that is what it takes exhibit was based on more generally if you ability than over each of those citizens of so if I a 0 2 of 4 an over each of them to take such a it's clear then namely if you 0 1 form you just have to focus so then there the small fizzles comefrombehind automorphic maps between the buildings and a new mayor true where tied to the fighting Muslims sometime that is being composed the composers to find the tangent of the bottom of American the fastest time in the match said and I things that conditions has not been given you but comes up so the ceremony for the soul so In this summit talks on the 1st of the underlying space just as as a country results more structure than we looked at all the relationships when I prison structural could say more about a proper just another box structure on this discuss that and person was structure on those 2 things and at that point you on the level they can just unleashed some upstart controversial results which brings things in general position that also structural support so here but the inheritor of structures which comes from the supposed stable maps so we had this input all this moderation maps the front of the unit the Of the plusminus well we just composers with this this is projection .period here and then the gets such moderation of 4 yeah the grading the grading the take from our the underlying object then can the composers and that is actually as before except that this thing now hasn't been more structure over each object says a structure of each object or objects that lies ahead of us space and the next thing is to lift the data from a story and the data from history means in particular on the cover business interests so How does it look like actually rather trivial sold in the base we have this chopping form .period and then you just over each of this
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parts you have 0 1 form just put it for all of that and it's linear it's it's 5 was a linear such as objects which is building in each of these flaws I have 0 1 form and shoppers here analysts take the fall and that is but isomorphism on fire so it's then satisfies the sources of relationships which we have had so that's that's basically completely on the most to think about it and then there's a functor revolve has given the chances of from buildings suggests for each map on each buildings coming from each building just applied the question so it's a plan and soon let's not think about you might have noticed looks really wrong Rosalind Wells economists but if that is the Jost ,comma a yellow means sort of that that 1 I want to know that some of them the redundancy and furthermore that I have said so is now it's 1st undivided discussed how what they can do with your soul so the hostility to to the idea to perturb Seidel Jr In the 3rd Jr I think it's more options here so if a part of Sunil Moffett object into this and this vanishes and this thing is gives 2 0 victory at 1 another was 0 so Sarandon's besides of this form so then the onetwo punch of theirs and what see here this gives just the raid warning for the 0 section so the idea now safe if if this is a 0 section locally I want to sort of have a petition of unity of this and just moved so I view was a 0 section and several 0 sections but was rational rates and moved this baby reports way to achieve transfer but it's more than what they ultimately wanted to so that's that's the idea of causing you run a so so this 1 room the pattern said but but but new moves the way you want to keep the symmetries should stay not for example locally selection of automorphism group this whole signature did so so it might be so this been turned into something like that if this is a sad if this is your sections for years and years of Houstonbased the and here's some of the it what sort of look like this and you want to turn it into this but you want that this whole storekeepers the listings in each of which fractional rate up new ones 2 1 that this is in there but of course you do the different places so so it becomes a little more masses off you go globally so there and then the sings a constructed that might be then but appeared and so on so perhaps so that is what you want to do so you need to develop some of machinery to be true pick such things will follow so that this up so it seemed if London 0 is replaced by someone which consists of this section then this is only positive if the question not operate a but such as section said it is 0 but what does it mean that if this is a golf course which is this puts the rate on a graph of different sections than this year only for perturbed ordered become positive if if you sort of course you have modified because 1 of those things in the gruff and this you want to achieve translucent and his assistants than that actually is was this was what is so that's sort of the idea of others and you have to develop a little bit of machinery or is that the clearance of but the aim is I you children removals and working with the possible he will have to a but then you would there was some symmetry and I think there's still a serious but it's not the cereal the it was us and I think you can actually do what some would some really good stuff and ignore songs some also structure so that it can do including that's nozzles that's not way to produce data and then all of this person who can produce something that's a very interesting I don't know it's like if you do as 1 in England lost region you just forget the fact that is as 1 environment and then you have some so on that level thinking of the and was not in way the same to something of an imminent system and no I have a wealth of the yacht soandso media about the eyes it is the LC have to keep them but the but you might not selling something when patentability ability warned that fits but you might be so but I think it could land could could could be better you have to think about you have to think about it because just saying all this should fit was that it is easier to say there's a whole lot of structure which we haven't actually saying I'd Aleksandr somewhat locally like I could say it should not be banned under the under the other other actions the isotropic group and then say convention of global because because a constructions on the local news to you so you rights as a seat this perturbation as the sum of a lot of perturbations in a constructive locally and then you have to transport them all over the place by the morphisms the mother of construction what but but it buzzing it's it's possible at least In a general framework to if you have a criterion and to forget some of structure put something you could definitely forget the structure that you want to know the new pattern that if you add tourism industry that they should just appeal as tourism office in 2 could use a new perturbations so that's would stay consistent as long as the environmental morphisms who could also disregard the fact that season just January I mean if you understand unions that if this is solutions those who could also go away from this however I don't sink for the little 1 you would get a new new series because since it can do it there's a musical what doesn't from from not doing it to the 1 and then I think you raise data might actually get just a lot of consolations but I haven't have characters so there's a lot of things you can think about what what so someone other requirements on Monday guarantees and resort properties for Saddam and of you know all this is that even better to formalize for
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so the 1st of all we have to define something before our products so if this on the fiber products sings a song I II canola don't write down that 0 9 a lot of what's 0 so then you just takes a product of the things I want to do the of the year also has a lot of things that yes I want everything so this 1 was the keyboard touched the wrong about cancel friend but so this is a restriction have discovering fungus so early on in the side he has guided by version of this which corresponds to the version the 1st lecture but this sinks can be also lined up according to the underlying faces that which was sort of a lecture part 2 was a lot of confusion which opened decreased in the discussion sessions of which would be the same former about sociology working there so this is the right hand side thinking about last breaking rights but yes this is if you have this problem throughout I mean it is if you look at this this is a case of this just what this says is basic foods materials should arrive before so what it says is that it has maintained that it was an interesting client and no just a lot of conservation solids and removes 1 term in view of this and the next thing I know there's always a lot of my life because he has always had a lot of nothing so now he has admitted that there that is not important so if I have an object of then it has associated agreements office ends then there some fares can be the composed theirs the company so Salesforce sink the building of hardwon got the parts of the surface which carry the things which are unknown and not true and the things which carry prison and if ever bidding the truth the burning of shows a lot of of guys and sold look at this and you can see the truth in the buildings and the rest of the country so that so that the natural decomposition and have a forgetful fund .period namely it forgets truce because you have done already so now now there is 1st of all it didn atop the composition of the so if I look at my head about space and have a 0 1 form of all over it I can for I can put this saying 0 0 0 on the tourists Linda battling over the truth building or I can put it 0 0 on the other hand the comfort of the compliment so this year's upon which which is defined on the original building but it is 0 over the truth component in this 1 this perhaps none 0 the chosen accompanied but 0 and the others company so you have this the composition of the student
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it was so you have a little village and bottle the new building has known buildings in the center of the level of the plot of all you have to do so at the end of it was the worst was quoted this is the case in history and the loss of millions or was about to contribute troops the it would be a tremendous buildings so so if I have something and then the producer said but I have some not nontrivial setting the viewing of Jerusalem and then I have to pay and also it would put soandso 0 1 from this it would so the Yankees he would just put the value here 0 now not not here but but it turns out be need to inductive steps for picture constructing longer than this .period bid already appeared earlier and then that is a single already having acquired properties here to begin with the something you can 0 in on the 2 a German amorphous so here is the picture so so I would say so so here's a lift so here is a list of the middle it's just forget the underlying triggers the land of the living and just restricts and gets its new object to Europe indeed yeah so before this forgot about policies didn't have no use a part of the book your objecting so so I have to find wonders is Pirate pie is just the projection
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year of his witness copper the composition on on the on this so it particularly preserves the underlying its it covers the identity when objects but that but this 1
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city does not cover the identity of colors see fullfrontal below richest always travels to use in the company what can I get that because of the if you want as if to you wanted to know what to do so if you think of this year as the preparation of producing ultimately data by integrating forms of components and so on then bring you there and then the next step is what can I do with a data cannot represented as a chain complex or something like that then if you want this thing perhaps certain properties and in this case the about property you have you have to go after discussion of the look of course completely trivial but since you may concatenation then you have you had something into it and so on so so the playaction untrue role and so that is why because so if you were disregards and somewhere was also some Sunseri but it would be different a possible so so now and this is of course a projection here and here if if the underlying object doesn't have to competence then actually you have this identity of the because there's nothing to to put 0 OK all for this fungus so this this is a country this is also a retraction notes many on the foibles because retraction of softened is a structure that you have entered into this series commute In this way and then what is important is but if you restrict the tool the new singleparty 5 of us another morphism but that's important that that allows me to put it back perturbations by this because of the linear linearity defined but the USM morphism before so so if you go through this list of things to you find is actually rather true in the concrete example but but this kind of things however arrived action all the problems that fuels the reasons such as always kind of structure what requirements so so this this pullback operation and he also knows about what does that say so the number you should satisfy soul if London is positive that has to be this service part of this has to be 1 and what does it mean this is this means there although the trip Rosalinda part of the component of the is 0 but that means is I actually don't pattern watches and no so if you think about if you don't put arbitrators analysts that means by and 0 0 0 the means addictions to cylinder now I don't have to put up said the clear so if Monday's positive view of London lovely hero than this 1 has to be 1 so this year a part of the overall a truce in the building and London 0 is all regional thing which puts rated 1 on 0 section and the other 1 0 so if this is what this means is this is the only that means there is no call for the overthrow of component is 0 so they did bebop hard on this if if the ball is because that the electricity in the it's security so that that is see how it already produces 1 of the properties all the time and the the picture of the world a little differently not because of the inductive Mitchelson so so it's something that on some level of like 11 1 building of course that is something of which would satisfy this property so whatever you you construct in the pagination because of this order isn't it it will actually not part of all of our troops and so you would always get Geronimo president of the British so then then of course there is some of what we had before or if I have to to Whitstable buildings and it and put them together a way I can move dangers of a new ones that property would take 1 of the representatives so now so what does it mean if this is that's just discussed what what what does it mean if if this is positive on an object so 1st of all it means there exists a rational numbers Sigma positive the vector in the fireball over the object so that land of use the In other satisfies this equation the missile raid Sigma nothing was the number associated to sort so now if if I thought is actually ability of height Capers once a top floor escape then the signal can be returned as a product of positive rational numbers in each of the other 5 Saturday and this year then of course there's a sequence from 0 through the case In each of those satisfies this equation there's a basic law we think we we see he also but if I so if you look at it so hot was interpretation of London composed so it means sins in the 5 of different factors that it satisfies 1 of the creations coming from the moment when the vectors was nonzero rates and they have a weight coming from the underlying form and so I get a sequence of a creations and each of them carries away the at the rates all of its 1 that's for sources of splitting what is your research to Ukraine itself but none of or not might not have not not just so so I to live in some sense become solutions but the dong condom 0 1 religious content is a rational rates and of course a also but but this contributes so so if I have to a solutions was wet it topologically account and 1 in the creation has weighed onehalf than the powerful thing I see is 1 1 half of the the creation because 1 of India not so it's a system over creations real if their creations true have a solution and just it contributes according to its right now is like the S & P and so called because a capitalization of company Thomson said but it was also interested in buying stocks and residents of this kind it also let go of me and has kept none of it yet so so this is of course what happens here the regions of somebody land of news segment of the Hooters so that means if I saw creation the body could this aggression Collins and is taken into the general bookkeeping was the rates so so then this 1 year but if there's a length as the composed Minnesota Member of the artists and we have this product structure and sold it's a question of the CIA has this way I can't put this together to this 1 and the signal comes from the individual rates of these parts so then because of this this property here each EI vanishes on the tourists and their components actually it's actually on every Allied because this 1 was already patrol and is the I know so this is this is the ability of hide 1 and this the I've already satisfies a property that overture 0 so so that so all other Tracinda bids for example this 1 here the blackboard all of this if there are
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solutions would actually be real Jayaram Mortensen so then if the offline has different components so on the on the only on the Seoul some on the on the floor then this then this conference you can decompose it according to the different components and some of them have truths in and each of them actually has a weight and it's a signal I would be a product of the rates for the individual ,comma so this is what the perturbation although some sort of individual companies of perturbed separately offered fewer tourists most all beechwood oral Moffat and so on and so on so now because To the smallest sold saw the body of you know itself clearer now you have to put a small structural Anderson and see that they can do to define what is was long on songs and then the the already for the perturbations inequities in the of and in and that was something which the union accommodation at all the people this know that Arkansas sold more of the following 1 of which is to say so I had a different understanding of so if I have to many false rape of beaches onehalf I couldn't and has fallen many falls by taking a proper vigilance rated onequarter that would be considered a career so then you can so than in the then if you have all warned that soaring fuel some many faults and use the money for those red what does it mean that fit together basically means that if it takes a certain number of copies he understood number of copies can match them up so that the rates of the so smooth flow should consider so that's the it 1 of the few said this is not it's in the middle of the Congress of the whole Center this is no natural identification just can identify after all ,comma love love love I mean you can put some of artificial structure wanted to say what you have to do at any given moment and we all want somebody who can say you put the starch on Thomas says you have to take so many copies and that you have to identify resist which is actually but you have to do when you do extension probably do extension properties like 4 sections the same thing for sections because when you look at how we can have a section defined over the boundary how do I extended to the interior of wall the only message make a local extension take a petition of unity but but if I don't know what to identify as wanted what do I had actually up there so you need that structure of this identification extend and then at the choruses identification together to the extension of the bone thank you could trust me what so it's another serious there exists and naturally style means up to some fixing some discreet set of data the strong bond of structural fall this year so it is pretty basically like support infrastructure for this except that you have a strong bond along a wall we have the Associated translation will so the objects firing so so so the objects on top of that vectors in a strong bond the average fiction by the All of us all property group on this and pure ,comma visavis this comes this isn't 1 of them sides from the pontiff obstructionist and you have this in a coherent way and then if you have 2 of those guys then began the transition said and that transition should bungalow over the transition said Europe and this has a strong bond structure as well as defined in 1 of the lectures will last so so this would be solved the generalization work of the structure which we have here and at this point you can start talking about this was not his section said so but it so so now the Texaco should fund the goes from here to the I'm actually just look at this composition in Europe and it allows in the original this 1 you get a lot representatives and it turns out it's as if at but which was defined by Katrina Strong and the moral i categories so that is so that this is 0 as a property the it's all space intersected with huge connected components In the underlying space used to have job always present this misconduct Branscomb of compact so scenario it in the was setting the caution 1 section is this action small section of but that property Enceladus is from then if if this the same vanishes this means Moffett objects such as some of those states associated more category to this 1 the figures as morphism classes and intersected which was a connected component Of the old its competence with political compared to bets by definition what it means is that you freedom formed all and as so as statements good but it will be toward more for 1 of the good things all that this fungus from the pontiff obstructions to the south In inject is other things which are in the in injectable form objects so if you pass orbit space and you get so there is a point view object here which is to the original given object out a form and so of we all realize they just taken important phone and wonderful so then this is a strong bond put for the levels and so that is what statement by the end of year last summer 1 the balanced statement we all have to hold on to all the charges and so on and so on I only goes so that means that given any object here there's a selection of set of those guys and you put that in there that some of some of the more construction use the object of the investment is down on their side qualified for U.S. used only once the soul so in this case and fought for this construction which you have if you take any of those guys it is the front on operate the sense as we have discussed so the pictures here but I described before Jefferson was fond of you gonna put my hand and then series of many forwards bonded over this here investors equestrian and fond of the trace what does here and there it's not true that actually reaffirm operate and that I can put it anywhere and and and this structure gives if I know something you can transported to commit to a neighbor of any isomorphic that so 100 so now he has appointed for packaging of the as of today so we have a strong bond of structural policy for the course month section functions assistant was for the SEC was covering fungus was compatibility and some additional staff there we have for each phase of I haven't defined as oneself clearly defined for on on the
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level of 4 errors also face he event this year's just the stuff of you lying always futile and this was a covering from then there was a number of compatibility conditions which were discussed in the last election nauseous and discussion love this tolerance of the things and so I II III or prayer suppressed Europe of the moving off but at the mouth of the continent's engage each other then then all of the state of anyone can have a bright on the treated before requirement for the perturbations you wanted but but that is basically solve smallest packaging of the date on which you need to do to produce a great opportunity for us to the last time I was in here is the August fighters commute I'm not going to be the 1st to you which it should talk about this will be a the ethnic conflict has been I change the sound level meter high up in the news no so such comparisons we should put the sections that put the course remand section during the mock representatives so here this isn't it so that it was checked structure course money Helios along the way I want sold solely on so that's good because of a restaurant and so so if you apply if you applied you also don't want to say know about this said this small controls this 1 so if I the Sea Hero is the identities of those not too much happening in this respect to the bar so that it is controlled by the sea but the CIA's certain properties with respect to the please when this will be 1 to send you a view that is the proposed that the bodies but identity known no no if you have a solution on the on the solution said would J. interested in his identity minds .period comports with the public 0 which means on the tourism industry would do would be to remove no identity minus part composers the body from 0 means on the true Salinas you happen to more flexible less said the question is I would to it Bloomberg then it would be what you have and want to say that is not the highlight of the Heller not noted it would not be the knowledge just said only on the solutions that so I think Yukon cement and you can write certain things under the dish under the sun so identity minus piety by 0 provided the these days at all provided actually the but the victory equal 0 means a lot that is the source of the Consent identity minds by composers the Bible 0 precisely means the both of this is the truth cylinders which associates were removed the letter that I had I wanted to do not Arkansas something that directions look so that's parade on this basis 0 . 1 and 0 so there's some truth to instruments have so now constructions of plus such section fungus a job particular class of guns so so this is an important covered some of the Simitis section fungus which you can sort of U.S. monitors sections of consider soft Compaq perturbations of so there's a lot of section is Of that ,comma particular kind provided has a following properties so if you take if you take some of this uniform marvels and the underlying saying the 2 0 0 would be the object of the real looking then this composition yeah but the columns of the number of industries 4 the 1st having the number of industries where this age far as the so Solid Gold into the bay not periods so age flowers In so age lies in a European flies in all the listings are defined on all and if a of the underlying basis .period is because victory put it Yukon the number of India's analysts and divide by the number of innocents who have and this thing CEO should be local supply sections and then we reminds you what that was this strong strong bond come this filtration maybe it makes sense to talk about it K and here and there those that can increase the 0 less and lost 1 so in particularly you have it came there you have it can 0 1 nite alone always 0 a the SE plus sections are actually going from here to here a lie in the fireball over in the end ,comma implausible not so so the insiders so this go from so that the final 0 but they're going to carry them and plus 1 and then of course but it is important that as well said his company should go this is a fiver regularity and if you view within the suspect of different moment that isn't complex inclusion the historical this contract provisions so this some kind of sections but there are constrained by having those properties in there I think you you talk about the world was mentioned in the the Wallstreet and that this has I here so there exists a fundamental science and export the center line and you look at the coincidences so basically the picture if this if this is all year and is a viable then locally slivers of numbers while Parliament is in each of the carriers of 8 1 0 well but 1 of a number of elements in this thing and you just look at this vector young How often on many of the How many graphs of In Richard the service vector subsided this isn't In the at this time of year and vowed that 5 of the rights locally in a new chance only 4 months on the soles of his multisector section from destroying each fiber the refined number of victims this rich having rates adding up to 1 so now if up the child the half this of course only image but this should a lot but this difference should landing on graphs softness across the no so so you put your hand in new see in the fight different .period lineup is lying on the ground of oversupply section so we want to and an end to this section should be self compatible with the group action hundreds of a compatibility so so there is the action of all automorphism group on the set and yes the Olmert's under consideration by the sea lions so let me 1st
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say certain properties what it can do is guy so you can build this summer the arbiters of a convolution and this is a small story of each of those Garzon in this season was arrested plots of repulsive so is as a the plaster and this is supplier because what is at a presentation local section structural missing you just have to sections such as all I for 1 and 2 yards for the RTG for the other you just take all possible that to things and just take away but as a way to take 1 0 while the number of the industry's twoyear terms in the community as a whole None of the other Zephyr on sun a convolution that's better because under section structure you take basically all possible Singson Sorokin adopts so so if if locally so if locally but the first one is given by spot and the other by side prior prime and this index set up in the excitement pride than the sun is given locally by taking all this combinations but his visit but rather by the index said is actually I cross platform the same class the drove so so then this 1 year well thus the place of the sections locally by teatime system so of course there is 1 of which on the other side of the otherwise so this is the most prominent so OK so if you 0 you just did but long as you puts longer the 0 here when was the last of all so it should be in there should be here it's it's the section that I Rich just as read 1 under 0 section I mean this is sum was procedures so if so that so this 1 here but does that mean the indicator function here it just means that the local structures to the times as and if she goes to 0 new goods the answer the 2 of 0 1 this year which is In view of the world that was the place was 1 His wife so that is a sections most firmly funeral changed so then if you haven't if you have an SEC was finetuning to all so it's clear what that means that music composer was a uniform others as his newest then you can put that in front of know so you can use petitions of unity to cut off such mergers sections then this makes sense as long as locally the new new year .period in uniform eyes or you just have to that the firm locally finances to take pawns and then you have only fundamentally 0 vectors and just at the mall options way so I think this is also again good section so the soul so this important facts lecture constructing perturbations this allows you to construct things locally and then just as things up and then but good factors if I if I give you any socialism was as small as object Anderson was rector of index 6 sections such a Londoner of as supplies refundable land his poems so I'm going to shorten hold proved so so now for example locally in the memo of encountering was describing the chances are that the results and perturbations on so hollow so if I if you have a fertile or even a finite dimensions if you have a section of the vectorborne and you want to make a difference was about small perturbation what to take is to add to its onto your times a perturbation to fill up the call ,comma then you sort of this the suspect as additional promotor get manifold and then project onto the parameters you edit undertake a regular value and for every article value that is a good part of British so now we what do we have local locally that what we want to achieve locally as we want to breaks a cemetery that is generally have to do to To achieve transferability of course we can work this then he takes the Audit Office perturbations also trends so and then we have maybe some or part of the nation's but for each of these problems it's precisely that already so when you look at this just have to make sure that 1 of the locker problems tonsils so get settle for a measure for the preparation the intersection and then you take some of the vendors so so that the only additional computation brother was used to precisely this so what that means is that actually rather than talking taking raucous sections to construct the local monitors sections and take that some individual commanders section depends on a few real chronicles the Texas right so that it fears aftercall ,comma and then among all these reasons and you get some of this branch many forms and they have a projection on TV and then for each piece of many 40 require the projections were that is a conduit conditions and so you find a regular absence but said that's the only at different so it's a straightforward come from there then you can even go further but for example then you'll know be have a boundary .period and the current analyzes of it's true but with respect to the boundary like its tank is it's tangential to the boundary then you could that it is if introduced larger sections of a particularly in areas ancient the connection tilt this Zaccardo into the manifold so but that is also a little bit so for this you only have to construct the section which takes enough values to fill optical Corcoran here you have to think about that it has opposed specific it might have saved value 0 there but it should have and particular derivative which together visiting hours questioning operator has itself said so that it has the same problem underfunded and so so there's nothing new and it is of course not so surprising because for them series locally the Final dementia problems find something you don't have to care about you and for the final dimension saying this perturbations of as which is in the final dimensions what so so let me just explain you how I construct the section so so I so I want to construct and on the floor in the neighborhood in someone to construct a section which has a certain property at the object of you you just let us tell the moderately so but not right and not let me I constantly on some level Nigerian precise level and still have 10 minutes nor would From what good so you know you just have to put something material on the grounds of an so but the side to side so I to construct something or at this object I thought was is a given season was always so it so I take such a uniform are so here's a picture the underlying thing in all its space that would the image of this 1 past orbit space would produce red stuff so this appoint somebody here which cosponsored the object taken neighborhood you so now no I consider so now what do I need so inhibits spaces you always have some was bombed functions but uncertain about forces as rather unfortunate abandonment forms of our spaces we don't so I think the otter in the north have smooth Bond functions but so also was a study certain 30 years ago pure interest in this was a lot of literature which Bonner was bombed functions on but this season was bombed functions are a little bit more there because as "quotation mark so but in any case the Nunavut spaces where available on set up you have to hours so so here is my heroes he said old and here is some of the neighborhood so now we're just here say this is a point which goes to the object of Fox Nora just Constructivism bond function this object to so
1:00:29
so you just take a bond function which at this objects corresponding to order followed Texas value 0 which corresponds to the given thing of it in the final the Texas support in the Small said and then you would take it around by the action so you know I have a lot of things so this is now on the image of sidebar of care so that then we did defined by this form along with a bit of resources definition and now the extended to the whole categories namely if you have any vector then if there is no if there is no morphism which brings the underlying basis .period into the into the image of old you just say it's it's modest section which has won on on the 0 section and if you actually can reach this page Europe then you just defined by this by what you reach and that is a sum was formed because if at all from 1 to the other side of was transitions we've another section structure judges can move forward so so that is another construction the note so now you can take a final number of those Palmer tries by Peter Phillips recall cornered from the front on property picture guarantees and nearby is also Zacharias it was a different sports car covering the complex solutions barriers and then you have enough things to do precisely what countries said some time ago so and so now generously don't 5 minutes an hour industry of stuff here in other words come on a you want to know the answer my view is that and and a I'm not saying which is to you look at so and this I sure wasn't so pounds portended that the underlying spaces and threw it is part compact so so I can look at size of old Dixie but associate watches and clocks which is sort of this red then that is here so then then in this other offers and classes is a clause of the original object of funding which now take neighborhood around and then then the thing is that so this is Emma tries of the space so it's so I can actually find a small enabled around that the clause the whole spaces that contain that's important because otherwise at singleaction becoming continues so then then you take the pretty image of In all which is proving so this red stuff at all and this is a pretty much of this you so in this 1 you take a bond function which has supported this and you you some is the porn which corresponds to the object of which was as the object of this category lies that is the object of a so here comes from a point which lies in the blue so blue region there's a spawned representing prosecutors 0 and all this other father was fiber there was a victory which corresponds to some vector online always Q 0 in the bond so now adjust take a bomb function which is 1 in the neighborhood where this point times this effect on an extended so that so I haven't thought about extension was announced but the what is on the part of whatever quantities so couldn't Austin on Friday and a control order so so so this so there is a section of support in the bruising and now you you want to construct the Farmington so what I do is I transport this section around but conjugation so and then I defined that nite then I get as many as of course some of it it could be viewed symmetric section and some of them are the same about the government of this is the new index set and you give each of them the rate 1 over the all of the group In view of the original so this is the head of the family still don't know so so so this is actually coming from the opposite from the Group of wasn't so this is so if it has a lot of the property but in general I would for example to construct something like this effort which is chief sometimes was honored to know bond then since the buyer is a font of 95 conjugate the Fondo doesn't change now but this 1 0 so that the perturbation but this 1 is the conjugation of the perturbed things by so it's also plans were also used the want so so removing this round doesn't destroy Arkansas vomited and then of course in general you might see some other sections coming from the overlaps also support but for the for the construction of a southern minimalistic thing you have to do and then you give each other the raid 1 the number of elements in the group so that's the size requirement so that means no on that smaller as the new look what happens the section is notifying them that since last summer after extended so then did I get an object here some vector if if there is a more fair if there is no more for this 1 which reaches a point which lies in Europe Group so if I cannot if so it if so if so I like reach slides or not if I can reach the slice by more fizzle than I would if I cannot reach a slice of put the raid on the 0 section 1 and if I can reach this by about morphism then underfunded like that so I don't want it what point is figure was the same ground and that is Cplus plans section notified globally so so it's in our it's so it's not very different images really always the moment constructions and then the length of time that the language of sort all highlevel that you're basically oversee everything on the knowledge you don't have to go on complicated corners and see what it actually means what you doing so that make that makes it of course according would be current series but but but but that the language there is much easier to stay on that high but it doesn't on the highlevel every information sale and abstract results will produce whatever you want but it's so so I ask you the thing I think about this in the sole what is exactly what is the reason that the team had only holds also working in the interests of local police and Maluku attend you you could put rechecked said she wanted but the guy counterpart and enough I mean that this is the largest saloons and what is locally modeled on retracts rather than taking just something which has 1 child about return that then could replaces 1 shot by the extra retract I this it also has some advantages do cover been overnight discuss a covering so what is the Malcolm Watson said some point of calls in the whole thing which has been suppressed I have to give a definition of what is actually a car in front the whole thing and then I to give a local model for then on the top generally have more points so it's actually more you win free trade it's going down to the so it's so the things I is about 1 .period bunkerbusting you would be unnecessarily restrictive To say but made it's a fair question ominous like that and they could and would be many forwards defined many Fortis sums image locally as charged isomorphic to an open said in any and I justified this is a metaphor is because law clearly offered to some many forward to assumption was made for the transitions movement they also show that the whole no no no no no no no no no no that was also once 1 side is a category and dozens of options some their funds news kind of object few
00:00
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03:07
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18:57
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24:51
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33:42
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50:58
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