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2/4 Polyfolds and the construction of Symplectic Field Theory
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Title  2/4 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  10.5446/16295 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  Topics: 1 Polyfold structures. 2 Consequences of polyfold structures. 3 Weighted categories and their smooth versions. 4 The polyfold of stable maps. 5 Bundle category 
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00:01
the has been a and on and on and thank you for this wonderful introduction I have had so recast but is so this was a category of stable maps so it comes as a grading is the memo Frost wants a loop year nutrition for the top floor then that there are some is stable maps that have all going in going ends so we have to In moderation and which funders then taking having the structure can build the flag the CFA Abbott products then there this sort of shopping from Dulsea should take the building was topfloor Foster was 1 thing you can shove in particular place in the funds for numbers in India the but this is a good tool buildings connected by an officer clearly the truth the building 7 . org never objects by themselves because they don't such as as stability condition but if you're object 1 to the more and then 1 can take a strong union of 2 objects and which produces a lot of fun at the mention of family of objects because it himself shift begins so so then this is what I call the more sophisticated Facebook funny us but in order to distinguished subcategories so the subcategory sources for this would be just all objects this is positive but each object would have to wait in the particular case of 0 and 1 9 0 wouldn't take this object and 1 what so that there we have we had that funds if it is in this category we can talk about Moffett object and that was solved in front of which takes millions 0 infrastructural more figured 1 of theirs and it had this property will be 1 to patrol of this Europe into some other form soft close by but of course we don't know yet what that means but wishes then and assorted subcategory resolve this thing is going to work responded should also have additional structures which have introduced during reflection so that that was 1 of the properties the Commission wanted perturbations you you have to have and this is satisfied in particular for this 1 but it so so this
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was so far larger bog there was nothing multiple injuries must so now so far have all declined to is survives the whole election but during the lecture having more and more properties but because of additional structures 1 can put on the objects In the structures that could turn out to be more this natural thing so for the CEO modest natural constructions after you fix on streets of the then it turns out then of course the neurons might depend on it but it take a different different set of data on the ,comma doesn't between 1 chance of deals and its are independent of the chances as well so I don't what a good would harshly for this thing so you see the you see portal category to use them topology so it consists of a group on a category that sort the MEPs was an example of this so by definition this means that between any 2 objects you have finally many more prisons and most fundamental pheasants it's morphism is neither morphism and in addition to this I require that sort so you have on the automorphisms and require that the isomorphism clause so so we've identified Moffett objects that could help will be a class is actually a set and on this said I want have a trials of apology so if you look at if you look at the state that sort look look for example at the income of but the moral aspects of you could make a complete a similar set of like this actually but if other players all across to be by the Socratic manifolds this I get a category of stable Maps program of written then you have the energy to objects and if then you take the clearance closets from other objects then that is a glass no it has a topology and affected comes from a form of the 2 apology to finals on other objects which to remove so you can imagine how that looks like this there so I didn't say that quality of table but the quality of the functions for which I can do analysis so that would be age 3 away from the politician it's remarks the written raise from the functions and at the punctured the fuel introduced all the nodes introduce the report according to 1 that in this former corn it's it's really was rates events the quality when you can analysis of the rates have to be adjusted to a spectral gaps associated to the plane Roberts so so then you can imagine what that it away from the North so we have a sequence of subsidies about hours morphism classes than you can places in this isomorphism classes so something should be limited and that you can use a limits saying by climbing at the nodes and so on and away from this that would converge industry and you have to say something has already pointed or lastly complicated near the notes will close by so I suppose it would have to take 1 billion won and would ask me what it 1 and 1 what is so successful so that's how I can take everything would greatly suite suite the Sofitel I want to work and about space petitions of unity and so on and then I need to learn and that interests you want Andrew domains To see 1 because in a consistent manner some of this new structure you have to actually to divide by and have to take some kind History the entries of the smallest elements present but it doesn't really matter no 1 wants sorry for the Journal from its German so now 1st Syrians often made all choices 1 can show that there's a natural may try topology not so there's no way we don't have any choices give a recipe out to construct the basis of topology of course the different person might constructed from basis but the assorted apologists say so it's not so always when you do mathematics incident you don't bathe in it and everything is natural and usually looks to me like the growing immigrant direction it's not I mean it was a lot they the Arbitrator theory of absolute souvenir defines their maps I have to say what quality of rooms and functions all but as but as said so and in fact if you would know what that means but the topology is there if you look at this subset generated by William Moffitt objects the newest apologies precisely the majority different fares have to say several schools and issue replaces category was their maps in in the commodity contracts the purpose of the visit soviet the considers were considered so version which is most of so that'll be processes and topology induced by the natural topology which of the I think you need to find and not really but it makes it so the action at the start of this project that we had a lot of energy so so we're all just a lot of pages about the decree heavy roots completely resistant so that that definitely you can't generalize series and then during the course of a single sum of money have inflicted mentioned solutions SEC newsgroups but I another done for this and applications but but the language is extremely flexible so but for contact Nelson it was discovered he was conceived yourself and other group actions on the status of songs composed moderates but it's some now so on and so this already introduced the last the CSO system this is temporary .period fought so this is all the neccessary to many foreigners bought Mondrian corners of 174 responded supposedly final Group a group of former President so that we
09:58
have an action by diffeomorphisms and associated consolation goal .period is a category that the objects are actually this points here and there is a morphisms although spare us the G is a group an amended report and they are being viewed as morphisms from Q to reduce start to so this there's also a category of missiles little direction the category that the morphisms on basically the group elements together with information of a starting point and the endpoints she stopped is the point of obtained under the action so I think the thing that that I mean then so what would be the CIA what would be the series here if you have if you have a the group sold so in this category so so you usually would look at some of the more interesting functions of farming tools going to the school and from this saying you could go by morphism somewhere soul but that's not going to say that this government yes needed I'm already running out of time and you can go Art I think we have to here yeah that's it but it just as a Category to bring underfoot and that I can compare novice said a uniform was enough so this is not an abstract category said medicinal properties like Stidham so good portal the reason the travel topology and orbits but from the class of the set of isomorphism classes so is sole the uniform a missiles a truck at at the object of fathers of the troubled you of a a front from this group or From from this category from such a category of the like this which has the following so it's injectable objects and fully face was so that means injected was clear for a fistful means of closure of 2 objects and the morphism set our goal to the morphism set off of the image forms budget of then if you assume that there's a there's a point which is rich enough to lose object you assume that the image of objects in the gold to isomorphism classes is open and you want to know require that if you pass to other morphism classes you get a homomorphism onto the open set so they have seen such things if you will go if if you're given a but all the the neighborhood of appointing all reforms through parliament was like this here as well so but here we just have this requirement of so soldiers of the picture so we have this enormous category it is not a said but so that most of morphisms go like this it's it's something like if you if you look at it the altered by the morphisms me with this as a group action now you want to take it over transverse was sliced risk the isomorphisms but you can't obviously in general because of isotropic also tried to be as good as possible transceivers of taking into account as property then he is the orbit space is a topological spaces of this here which constitutes the the other morphism classes of associated to this thing here and you have this year into this category so the automorphisms goes on in this direction fled the here with soften collapsed arising from automorphisms closest to the city so it's something and so we have to talk about compatibility of his optimism this so if you have a lot of other more effusive in this direction soul so the differential geometry of this category something like that so I put a uniform and look at the trace of a functor no I was more structural if I see many for the 1st ever subcategory and it hits my hand in many faults was already some structure put this hand in Osseo also many forwards and if there's a morphism from here to here shares of Coronets which explains what assume this picture on that picture so growth source and you have to do that but so too was going to say what would be the ideal was new structure in this country To someone doesn't like the study of fungal on this like all Facebook from too much this was Facebook from so I suppose it's a is the original 1 0 0 1 so it like some objects and doesn't like some so let's assume in this room the other morphism school lost in this direction and put the uniform wasn't so so now I look at the both the light objects on my head and suppose they lined up as a metaphor Sosa pieces of many forms and if I put my hand in here was effusive many so it's assume that when will 1 object you often of a private beach up some related but has no office then what you want to have is of a change of from here to here which explains what in this picture to picture and you can formalize a lot of things like you can get conditions on this front of that what you see here is actually and is a topological stops there is a Compaq and on the smaller structure here that Harrison was structure under the metaphor to capture formalized everything on the language of fantasizes is rather easy and then some then there's some abstract steerage treasure that a lot of interesting objects you can create not so that actually In the approach he work on level the language as easy and there's a certain model additional infrastructure summit which chose conveys a constructive and wonder 1 all of this but just because it is abstract business the rather than proving sings about what our spaces that each point you have to work very hard because you're on the wrong sir but history so this is what I'm explaining now this was just so preview in Kansas where he last fall out of love OK yes I have not explained of course I could say if if if I have a subcategory and and look at the trace of this I could say it looks was enormous respect for that shot because justice the objects and on Sunday but was after complimenting the consensus is that the it some
16:58
support surgery from others that have been built the Big Five a product which is the following you go risk from from all you go into this category at best in the governor's the other into this category and if you look at image .period here image .period Healy assume they're connected to look at all also pairs of points which are connected by an isomorphism no was follows the prime so that the image and the is connected to the middle of a prominent supporter of of because of this and I I think this is you know how their colleges in a hearing on the of the economy however I mean sometimes referred reprisals would also become a problem categories I'm coming from a different planet but it's so so this year because the transition said cleanup of course what we would have to know all of this is a smooth structure and so yeah let me point out to you would be what structural things so
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if you have this you can go to if you look at the state can
18:17
take the projection was the European takes a projection was prime so protection of the
18:23
1st thing we called Salsman is always rather standard things in group was also the target map then a there's a there's a map which associates to object justice took place here this is the 1 met the 1 from the 1000 officer at the image of you you have an inversion not just in the chains are already earned just places by the universe they have been here this is the real fiber product was automorphisms in the source and the target so you require that the the target of that 1 is the source of this 1 so connection composers which means this picture can decompose but petitioner and this matter so this is Bayan is nothing gives us the owners thing the on interesting things you can do and the maps about 4 structure
19:22
so now here's what 1 . 4 structure for business so it's it's a it's a construct sold in real terms the former all this kind of categories which appear in real life the object in this category are usually are dramatic Although objects and have enough geometry to build something like if you do Dylan Mumford we could look at the category of stable agreements surfaces and just take 1 object and thereby growing by cutting and pasting going you a construct the neighborhood In the isomorphism so so the point is that this categories which accounts about the geometry for example but all of us that the objects are not just only points but there dramatic objects and then you can actually giving general recipes is a take an object do this do that Bob Loblaw there might be several choices in war but at the end of the day for each object all constructions that can do so here the purpose of so that there would be a fighter from seem said that the because that but which associates to object to the fact that there is a sense of such uniform Muslims not so for each object the the construction of a set of such a from lot then then if you have a isomorphism between 2 thirds of the together by ejection between the construction of the construction there so in general you can say a lot about it so this so it's always there a clearance of constructions if no 1 objects into the other than uses data that you choose your was morphism corresponds precise set of data on this 1 so so this in general application like Steven maps everything this by objection can be very precisely explained and then the other then and there the construction at associates to tools such as uniform was detained and peripheral for structural this transition said so that these things ought to look like this source and target unaltered this diffeomorphisms the wanted map there is a small was ziti inversion that business he diffeomorphisms and modifications so I think of this as a maximum recipe at last for this category so I so sold for grumbled that the 1st of 2 fathers there's a set of ruins rotted maybe as a page what you have to do the opposite with dozens of listening to get on and then usually this this comes out implicit funded construction so so that so that in general so the hard part is that should could In this in this concrete case is to define the F and come up with rights of conditions so now if you have this so this why is that it was not because of the the goes on Leicester Udinese so this is just inversion so the so have 1 slice you and 1 here and have a more morphism In this poor adherence to to follow his lead and this part of the problem and this fall and that's 1 of the important conditions so they s this would never missed was did this year's but and this would be mapped the crime and then what does that mean until August from the diffeomorphisms adjustments if you start moving this point here is a source would move from this endpoint would move was so if you have a slice and I find would cut out many for whom there are few and other more often than actually I could knew this object takes his other morphism and with other more diplomatically identified as a many for so so now but so that's a pretty amazing just having this definition what it can do we can write a whole book about his like but the new right when you introduce the notion of a menu for the can write books about so fittest idea you can do as well and like you write books about it because does implicit debt of constructions you can do on that level all going to local port of call me do was approved you have to laughter got a backtobasics for something but you can do a lot of constructions and of course as a singular is happy the new study a concrete problems at that usually this is a motion of the moment we have to get their hands start and then who creep up you go up you come to some level because it applies Sunseri which means there's a lot of implicit construction given constructions you can apply and then usually sings progress by Foster so it's part it's principally here as well but for this it is the basic notions of let's see what we can do with so let me 1st
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say for example which many comes up later when not if you have such a seamless structure defines that's the Nautilus was fractionally actually if you category so so I felt so so so it is such as such a structure that has attention to the construction for example you can associate with such a category 10 you can associate then you can take for sounds of the standard and the kind of look at particular Fonda's which December differential forms so what we saw the excitement of since then the then it turns out to such side and as a teacher saw no because independent space of all into the tendon category and then of course you can both put back such a fund I invented looks lot opponents was scene was set and and because but you only have you don't have to check it for each object you just have to check it for an isomorphism class of all of the front if it since I took this seriously and to a certain number of things so it turns out that in the case of stable maps you have the evaluation find that it Edmark points if you put back a differential form virtually differential from on the stock to drop compatible with the usual stuff so so you certain things which are needed also important there is a degenerative fund so if have this new structure on this cat part of a sort as degenerative form dropped name is justified the generosity of the object was thought to be that of the object represented that's independent because the DeGeneres is being processed by the transition that's a fact if I take told representatives the name of the same year and episodes of the source font and then the objects have 11 structures that can say for example offers little carry the can write like this and that lies action OK on the Ltd so it inherits everything yourself on the former no let's talk about the scene was versions of subcategories so so here a nicelooking portfolios square so so let's just say what consider Boggs submanifolds in this case but the picture the same general so general position would be something like this if if it's a onedimensional shouldn't get in the corner of the position from bonded good position about that still OK you to come out like this but of course it's not general position because this is 1 images should his mother corner but you might have stupid things like this spot that there might still be retracted so this all possibilities the 474 so now the general definition is the following so it was important for the subset then the the have heard what of Subpart if all of this this is a subset is locally regionally but now require more but acquired the retread subset if I if the index by 1 year but this is system for doesn't mean this means zeal differential ability property is stronger directions if you go from here to you you really have a season wasn't if I go 1 up but it was shifting the export 124 introverted byelection dividing by small sorts different abilities a stronger condition and it turns out that this this if you will however said the shock is a redirect of something which has this property then facilites ,comma 74 and for good reason namely it in the a new structure from the .period on it is a stop order for structure which however hasn't hasn't occurrence set of charts which are actually use of many .period I don't reflect the so what lies a final dimensions so what so 1st of all if our goes for you to you 1 you 1 goes to a new tool goes into free and so on and our composers eyes off so the image lies a new Infiniti so such retracts was 2nd only to retract onto skewers .period so then many artists muse .period so you just the 1 that you can take the tension above all and that at each person was appointed attendance pesos attendant of all on that sir tangent space is of course the identity so tee off on point this is identity but because of this 1 incompetent cruising the identity this compact that ended his Compaq has befriended dimensions so if you have this year you will see that all the tendon space compact so there would exercise like this so that we will takes tangent space and actually nearby you can buy the present but was the function surrendered solutions space as graph attendance patterns so this you already know evaded suck it was fired on in 2 0 1 which associates and object rational number in this range the support for the support of the objects we want but their farmers in restricted function which is way to know what is what is the
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smallest version of the the osmosis rated local phone division subcategories which is awaited subcategory that so the following boardcertified take other morphism class and but picking off I thought it was in that other morphism class and take take such a uniform was on which meant that . 2 0 2 0 0 4 then you can find the open neighborhood around this and properly verified emissions submanifolds here so that this fund written this way so it's an indication so if this is positive it means it lies on the source of many followed In The number of occurrences where that is each each man each of those many false calls upon all while the number of elements in feminine and it this just Collins on how many many faults divided by the number of many foreign to her you said so for example if if you don't fix the values they all want you would just see 194 and it was just be wonderful and followed and otherwise it would truth In progress was made in the don't dissident voices in general not like so then of course you can have different flavors like locally twodimensional therefore the same dimension general position to the boundary in good position to the boundary this thing could be tamed Oriental all we oriented and Sorensen of all these different flavors from 6 so do some exercises how can consolidate those same workers want to know but so then if this is oriented the differential form which originally was to connection integrate this all the and how to do it as well did this interpretation of the stuff about by data on the orbit space In end and and the differential form defines you signed measure on the orbits spaces and you and you have to wait on the orbit space and adjust integrated the way to the suspect was assigned it turns out of stocks here in this country the things that you have this region of of the world we live in a world well you have just as bits and pieces like normal and so it was that example then you see how that would work so the refers gives a picture so in local corner and was flying around here this is your soul and all you have the option of you that you have all this many lying here and there is some of the local picture you see if you have a thing was found alive visavis a picture you see if you stick your hand when look on my hand as he a green stuff he said I was in the service of the Reds starters enabled your cue and always the marriage myself so stick my hand and is the green stuff rustic man you see some other green stuff and if there's something other Moffatt than this s fact that this in structure as antiAmerican diffeomorphisms it maps .period here differ fitted to the picture the shortest In this picture however you might for example it could be so if you have a certain number of manifolds during this I could for example take another copy of each many forward just 3 degrees of rate by 1 by about half that would also be representation in its so some exercising remarks so I assume you have a small subcategory like this and as you the the only assumes values 0 and 1 between 2 objects and even to object to such a subcategory that is in this sort of talk read those ability those positive there at most 1 the unit to objects and further assume that all objects have been generous is 0 so intense and the interior so that I don't have to talk about boundary show that the support of this is natural a was company forms of bonds so so what you have is that the orbits spaces Compaq you have these local pictures you have a diffeomorphisms and adjust through this stuff together milk picture so that's so now let's you the because the generous is 0 so that always young so so if I wouldn't say this is what would happen if I get a submanifolds and in the sense of the difference in Japan which might however have bomber 6 but but you would have tended spaces at the at the ball at at this metaphor that the boundary if they would like good position to the boundary then actually can show that this will be manifold is pondering corn smoke it's it's the my but in this 1 and that was it I or good condition just basically means that somehow got so the definition in general the more complicated but that attendance space doesn't like like this and like just goes off nicely and that if you make a little bit of withstood monster it is this is this is 1 of the possible possible things you can require smaller saying just means you have this sum was representation life like that but then you could start to climb all kinds of things of Yemeni for like Oriental on having alters the same dimension on the boundary line good condition to the boundary line in general condition to the opposition to the boundary and so took so this guy's occurring all kinds of different flavors and he said he 2 of them so if if if everything is so severe as to their individual objects as most 1 morphism and the orbit spaces compact then this thing has an actress was mainly structure news from the from the start .period reported .period from category if I been required that again I declare that we all have only values 0 and 1 but the but assumes that I don't say anything about how many morphisms I have into objects managers supporters this was comparable so so services language and Toulouse possible things happen so describes many and all forms and then the most general saying is that you can have rational waits and looks what ocelots and generalizations so there's 1 all the more general notion and who specialize the car additional properties that are produced since that you know but but the horror formalism is actually much closer to to the constructions you do if you want to construct a model aspects of the level on which I told myself streamlined for saying the least amount necessary to construct something and then everything as you say so abstract stuff you can do "quotation mark and In lots of sun along rejection book about all this starts in construction that for each indicates case from your new fraction to do so can integrated archival so so basically the construction moralize space comes up you have to good good recipes I didn't object there has to be good repressed recipe what to do with this object a local construction and and as you see them and we also have to introduce Bongo's all of this and then we can talk about sequestering monthlong talks and then the nite extended was notices this vivisection perturbation theory for font for certain classes of font is already said so those attending construction and then but if this is going to amateur former soap so this doesn't want to see if you will construct the resistance was among I spaces in the and what of the orientations to integrate forms of others but it also tells you that if if you if you view on things from this general perspective so if you want to do as if to use your differential forms which metal would be put directs all forms by the operation of the market .period from getting the map and collapsing unstable structure going to the demand for space so this operation should turn out to be some with fountains and then if you put this back this would be particularly type of this differential forms but there should be many many others which perhaps some of the topology of a function spaces and said then did they tell you that doesn't have so much compatibilities so the company representatives as if to you
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but the muttered presentable than in some other way ahead of him you I don't have interpretations of what this sort of just exists so and and I think that my feeling is that many many forms and the just ever used and must have to do something was so if service and look at all maps into amended and ended year the vital but being isomorphic saying so its use on some level I think the the functions of the function spaces from agreements offices into into the many and there's there's not apology so but but but but if you use of such forms I think they need to restrict In what kind of injuries can get so if you if you put that other forms of model vary a lot of symmetries and so on but you wouldn't but but will focus but I think there should be other series as well you know you can think of no use to me the United States clients I can't say so I was going to the work of art that the question of the security arrangements as is in the artisans of that being said the government was required hold Will you only 1 of the year in which the 2 but after going young so the United Front results you know homotopy instead things can happen so the students to construct usually not Everest most of the camp's singularities and largest sensitive to salt away and British can distinguish and so on so serious as a national trousers topology renew instruments and then has a natural part structure after fixing and 1 its it's a minimal amount of discrete data but the construction sold General actually can use the same construction and fix the minimum amount of data from the item so whatever comes out of doesn't depend on I would think that this but that's not so bad as effusively Fobaproa which introduced most tend to lose the quality that you believe and also has a new structure and and then again the nice things so this national construction firms with structure the niceties of course the sunken reality check the generous index from the port infrastructure agrees with the great informed so that its knowledge of degrading founder was a highlight of the building minus 1 so now I was structure have agreed in front of a coming from that another directions some so structure takes very hard into confidence of this quantity on genetically so not now immediacy now having this topology of structure so at the office space they can talk about the connected components in the office spaces on 0 became talk about connector conference there this year's just the obvious things of that then if connected company and Abbas associated category for this nearly all all objects which isomorphic to an element in a the other mushers class lines then now faces In each other ,comma notorious not comes from this in as of so the fails to take a connector confidence in that topological spaces this but in the sub space ready because 1 so you just take care of honest boundary points now the take buildings of Hyde's tools to top floors 1 and it takes a closure of the West so here's an example in dimension Inc differential moment to solidify involved this year 2 square then I have would have to take this point 1st away then this year will be connected component of because 1 on Bondra .period that is a closure when I get there including the support he has another example is another connection component takes a close look I get that .period again so get 4 faces in this case no whatever those portions and the number of all was lives at a given point is actually the generous so then Singapore this that and then you get 3 phases come together interior points here clauses opulence of this sector here front a fact that and a generous the generously index as precise as the number of cases you let live the interior .period abysmal friends 0 on responded .period 1 "quotation mark singers say those in the corner .period line to freeze you have all on the last year but that wouldn't be allowed the of in that year but it would do so I don't know that in my structure here because of of his corners always but but we had a discussion of this loss you could rather than taking problems partial currency could I think replaced by Convex at this moment in 2 years so then you could generalize somewhat saying particularly lucrative colleague bond reasons I think what we need and Bloomberg the rest of those monsters such a subway given the connector companies associated subcategory nearly all objects was isomorphism class belongs to it most of they said was knows that given phases sources carriers so interface lies in a it's a subcategory of as so if this is a then this could be a but as a result of subcategory of that thing that would be the closing here and there's a lot of things he used to state where and realization was that the main it's so each sold locally because itself goes and this discrete categories is a product "quotation mark a face in you want to fix your role as saying that he is the the justices appears a urinal Fobaproa songs at least referred to and this is a topic is also has all you just forgot the market a apology as well this is a building right by the you have that many private life said yes because you could show should 521 crucial different places the children once the connection but you can shop several times and laws among the problem so so it could take a building of Lane's 25 In a chop it into 5 billions of lines for fight for a full pressure for me asserted so so so that the sources said that Israel the compatibility comes from the stable think could shop in different ways which lies some sense belong on the part of but it takes a different combinations but it should be the same perturbations on the next and so case don't have the same lesson has you have to wait until you're in the new students but it was only of the this
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His name was read tho he did the hero the UK and how it said so but so not at this point so here what could happen is I think that that is what I want to say this but if I have something like this that bank that would be the face directions but but in Arkansas at this point but but but but it turned out so so locally so far up but it's what I said before all of the generously indexes number faces life in Berlin is only true this notice now so here he is actually 1 faced but if you give local election 2 the students and the U.S. theater in this In another source of mobile so so then it turns out that that pictured you never run arises in its system this the also us that it has hated his main role would so so it contained so if if you giving any unbroken bill you take sort of all the limits of Canton it is not the same as they were known so that victory of the whole structure comes from but has a has boundaries connected which consists of a broken things and SB has also this but they have self common things also known as were once what did you think of something like this 1 beside around like it's not so it does not happen .period but in in this case for the for the music losses for the year in the relative case and not 1 but here in this good in this case but there's no so now in all of a duplicate is actually the faces those containing a pawn if you you can see like this is this is a building in this hour the periodic always think of his parents was out the stands for the top priority for the collection of popular recalled what Auburn's 1st interface 2nd of insurgent so then then this field things which contained this is a case and the close of so if you have this stuff degenerated might degenerate into something and then if you break a little bit higher was the same breaking of the city yeah because in our series there's more energy resorted to this all Lesko there's different complexity if I go operator laughter so I measure of the rate of bronze cylinders and saunas and complexity of some interviewed soldiers was so so that so few of these other failures across all of these things the faces that will contain here and if I end up all of them in such a way that if I break at the lowest saying the 1st phase of a break answer seconds or so but takes a clause of this it's smaller than that for it's a small country it the of the same now they're all in the same space where that lies and their different spaces but they intersect on that specimens but but but but the phase in our case is just give all going in going ends and red should break this set of parents described that's face recipes and that ended yes this breaks High also it has this thing has more energy on the other the last piece has energy this is the I don't this 1 this idea owner of commemoration not so so so you haven't already on but it stopped so if start now with the phase then then look at the map ever for the interior so it interior on the interior of the state of course nothing during the whole space this the unbroken string is is 1 broken so you can so you can actually you was of founder of 0 0 agenda was the 1st lecture to go from year to year to cut this saying which has 1 interface precisely at the end of things so you get bitten of high tool but if you're in the SC does this is the generosity 1 this means each saying has hired to tools topped Troyes 1 it was 1 and you can only Chopard that 1 point In the middle so and the chopping was if 0 was arrested at this the ministerial Minnesota's studies topology and so this madness continues on all the experience of past overspent his continues and it has a continuous extension to the closure the many of the pictures I want to be financed the way exactly what we're about well there did this is the flow of whatever series comes from the interplay of the 5 in which I explained and I think I was as explained in the 1st major the covering fountains associated with being the boundary of errors in the final report so I I'm using all of them and show that she lined up continuously by doing this can be creating this kind of thing so it's it's a continuation of this discussion by putting all the topology and so while I can follow if there would be no continuity when defining the weather was told him where I have where there were just so so how how effect will be on many forward and they were just consist of pawns and would not be talk about curvature so so I have to have a notion of topology and snow was news to be able to talk about so I'm explained to you now that his current form does actually are continues it's 0 continuance of space and has a continuous extensions that that says that you can extend but the thing about the introduction of the wedding will the result that is why we are here as I was implicitly understood but how how it is I am In the elections number 2 the way I want you by explained to you precisely that among these conditions under Bartley DE because ever starts Utah peace that is that the stuff comes from the Latin word for what he's doing this now this already knowledgeable Barkley but I have to put up this fountains I need more structure to be able to do so so no I'm starting this relationship and show that certain things are continues even the moods of the available to you content so this is a country as extension so the the image of a connected ,comma of any NEC a connected component can that space in this fiber product in the office space of this former .period so but also every connected confidence comes from a famous so so you see already topological data connected companies and the fiber product cosponsor faces and so ends here explanation for how this extension this extension of the continues extension of this comes by putting several of the previously eligible by defying things together not so if if I am on the if I on the interior of a phase the only find the structure out before can apply as F 0 0 then I go to the program than about as closure this 1 broken since and there might be many the sings a job very often broken In the obviously has no is no it's not that difficult but you have to why don't apologies and spent and look at the pictures history this was just the thing that there is still putting you in primary victories status as it was the
57:42
right thing but if you if you just would look at the 0 0 onedimensional complements other than its own trebled since I started discussing also highdimensional moments they will have the kind of thing you know so so if the 3 times broken life if 3 times broken thing lies in the closure of a specific phase so there's a lot of over 1 times broken saying Richburg was certain things then I have a continues extension of that fight on all its space over there the end the extension depends on which Phase ,comma that is where the OJ remark comes from so if I extend that phase is about to extension if I do this so L so here so you have a seat goes on as he does 0 justifying 2 0 0 but on the boundary it is actually looks like this this or that the guys the other box guys that useful so this continues extension can be represented as a front and pieced together from the air by James Michener does it needs so the FIG Jaiswal defined and certain strong autonomy on unsafe which had agreed to I plus J. plus 1 times but this I had besides you might like to yes and Vijay was this was just forgetting just for getting at the right spot the tilted markets in the world is something I will have a history of using DNA a pain in the U.S. seniors including in the area around the uh operation and fire so sued the 1st time I just was talking about the algebra of things now he eyed us and this was in some sense like if I J was defined in a completely different strata of the country when something sales order so now what this says is that you can't use them continuously together that's what the end the interesting thing is that this fountain of which goes from the category of faces tool this fiber the progress to affect us all over the company the sources so the image of the there agreement that company is so this fountain businesses was for cover in front of Seoul to have a small structure and all a small structure fall the final product and I can look at then it turns out that this induces a similar structure on this as well and then this thing is actually snooze function and Canada also was founded almost like this continuity said Minister can if I have an object here an object here which projects on it I can take 2 things so that that foundation the slides into this lies and the local presentations systems I think we have a lot of confidence in the new year said an example of the kind that has some of the things we already have something to show what I would like to find that the phrase and here I have 4 the Continental in the sense that there is it is not 1 of them is so bids it just says that in this family of 4 justified on different strata just line up to something continues thing and sometimes was wanted to but yet a picture to explain it to you what is so much so so consider this object and so it has patriotic assistance for Peru recalled its union it breaks twice but so this so if I look at this kind of thing here which would not break here that would be in the face and if I would look at this in Europe and will break here would be another phase in this Red staff lies in the quality of the lies and the intersection of his preferences so if it takes a close of things like this This clearly lies in it takes a clause of this that lies in that as well so was so now he already that's really different connected compliments inorbit spares come into play this 1 that 1 and that once of a B and C. now you also have this 1 comes into play which is the connected confident you and this 1 comes connected confidently and all this stuff like interconnected companies In this year of the no it but it's signal 1 of the things face and so this all connected components not EQ P C D and so now if you look the categories to go 1 in the 2nd seed through that's occurred resorted to the center of the phase that and you take this fund tell 1 then it would so what it was just Del Gold it would press service this year and it was chopped up this 1 so you're going to ask any then you get this thing but this saying here is actually a phased in N this is the phasein sq so it is a skill he than this 1 would like you but it would be broken once a would in sq yeah so if would look so this object lesson in the sections placing much of the world he writes following the right so 2nd segment tool 6 months so so this thing here on this this this thing lies in the face of Sigma Q "quotation mark as significant "quotation mark as signal to what is the difference between the 2 is signatories of phase in Cuba His work with their parents have always known and unknown I know this so but I would stop here so the connected component like this is seen connector ,comma containing this is called the in correct 8 the connector company calling this a secure the opening competent book containing this 1 is the answer so now if I had if I have an object in view of this this book is a stub of broken things so now if you tell 1 that is the lowest 1 would forget Daytona you would forget how can you get 1 thing which is in the same signaled told segment to what is a phase is a phase in here he said this is the real thing alive in the in the 1st half of the applicant at the same time as the secondleading will at last have the agency in May although it was a fan of the death of so here you would take such an object forget to do indeed young men you then you get this singular flies in the connector company and you get this broken saying you're lying in the connected company and Q but since it's broken and it has to be lying in the face so now if I do the same here forget this year that will be going downhill I would I would get this singular lying phase signal 1 all PE and I would get something in the connector components then of course I can shop again in the same and then ending up in the Triple Farm report and that's commuters and is also was covered from the World Bank and this was covered from that that that that's important thing so this actually poses more structured Wilson was altered the car in front of and then when you put that data released was so that is how all the whole stuff starts all over again he into the time no so the also of course and other bodies so obvious already itself with some something covering but that that is actually compatible with a growing and so on and so the nicer and nicer compatible on motion was in vain at action and I'm done with it but it struck out on his own that's pretty good so in at my so 1 these on the slide and
1:06:39
it led to see that there is something like sublime would have just say just a summer so so so it if so what want to see is some of the information you want to have was fond of us in the sense explained that that should satisfy this identity and this identity here is this the Ojha identity reorganized but now the ingredients to see Yeltsin was found any again that is why already this type of fun in the 1st lecture is the 1 to find such things which are most satisfying this identity and having all property and this seems constructed as a solution of the
00:00
Building
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02:56
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16:56
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18:14
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19:22
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30:58
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Glattheit <Mathematik>
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INTEGRAL
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48:46
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1:06:34
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