4/4 Polyfolds and the construction of Symplectic Field Theory
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Formal Metadata
Title 
4/4 Polyfolds and the construction of Symplectic Field Theory

Title of Series  
Number of Parts 
36

Author 

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. 
Identifiers 

Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

Content Metadata
Subject Area  
Abstract 
Topics: 1 Perturbation algorithm in the homogeneous case. 2 Perturbation algorithm for relating two perturbations. 3 Remarks on orientations. 4 Representation theory and SFT.

00:00
Group action
Building
Musical ensemble
Curvature
Multiplication sign
Sheaf (mathematics)
Kontraktion <Mathematik>
Complete metric space
Mereology
Maxima and minima
Heegaard splitting
Manysorted logic
Streamlines, streaklines, and pathlines
Length
Position operator
Social class
Stability theory
Compact space
Logarithm
Product (category theory)
Spacetime
Closed set
Infinity
Zeitdilatation
Perturbation theory
Sequence
Category of being
Cylinder (geometry)
Isomorphism
Hausdorff dimension
Order (biology)
Uniform space
Resultant
Directed graph
Point (geometry)
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Rule of inference
Number
Morphismus
Frequency
Term (mathematics)
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Energy level
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Algebraic structure
Division (mathematics)
Set (mathematics)
Convex set
Dependent and independent variables
Finite set
Object (grammar)
Fiber bundle
Local ring
08:40
Group action
Building
Confidence interval
State of matter
Multiplication sign
Sheaf (mathematics)
Hierarchy
Water vapor
Curve
Mereology
Subset
Summation
Maxima and minima
Group representation
Manysorted logic
Negative number
Moving average
Cuboid
Fiber (mathematics)
Stability theory
Social class
Compact space
Spacetime
Product (category theory)
Moment (mathematics)
Interior (topology)
Perturbation theory
Price index
Measurement
Functional (mathematics)
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Category of being
Cylinder (geometry)
Isomorphism
Funktor
Order (biology)
Phase transition
Theorem
Configuration space
Normal (geometry)
Convex hull
Resultant
Pole (complex analysis)
Directed graph
Relief
Point (geometry)
Neighbourhood (graph theory)
Connectivity (graph theory)
Student's ttest
Regular graph
Open set
Force
Centralizer and normalizer
Latent heat
Thermodynamisches System
Arithmetic mean
Operator (mathematics)
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Energy level
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Selectivity (electronic)
Lie group
Absolute value
Units of measurement
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Addition
Twin prime
Neighbourhood (graph theory)
Projective plane
Algebraic structure
Reflexive space
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Line (geometry)
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Affine space
Approximation
Anamorphosis
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Charge carrier
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Family
Local ring
25:59
Group action
Building
Sweep line algorithm
Observational study
Connectivity (graph theory)
Decision theory
Kontraktion <Mathematik>
Student's ttest
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Weight
Explosion
Maxima and minima
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Invariant (mathematics)
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Lecture/Conference
Flag
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Process (computing)
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Product (category theory)
Forcing (mathematics)
Neighbourhood (graph theory)
Reflexive space
Line (geometry)
Perturbation theory
Incidence algebra
Functional (mathematics)
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Sequence
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Category of being
Cylinder (geometry)
Exterior algebra
Algebraic closure
Network topology
Order (biology)
Dependent and independent variables
Hill differential equation
Game theory
Bounded variation
Limit of a function
Inductive reasoning
35:01
Point (geometry)
Compact space
Product (category theory)
Spacetime
State of matter
Sheaf (mathematics)
Survival analysis
Insertion loss
Set (mathematics)
Perturbation theory
Mereology
Functional (mathematics)
Estimator
Time domain
Category of being
Maxima and minima
Computer animation
Isomorphism
Theorem
Object (grammar)
Invariant (mathematics)
Subtraction
Social class
Compact space
39:11
Point (geometry)
Multiplication sign
Algebraic structure
Sheaf (mathematics)
1 (number)
Parameter (computer programming)
Mereology
Time domain
Derivation (linguistics)
Invariant (mathematics)
Object (grammar)
Energy level
Boundary value problem
Local ring
Category of being
Position operator
Family
Compact space
Addition
Compact space
Product (category theory)
Spacetime
Scaling (geometry)
Neighbourhood (graph theory)
Physical law
Dimensional analysis
Algebraic structure
Mathematical model
Perturbation theory
Line (geometry)
Category of being
Computer animation
Hausdorff dimension
Theorem
Dependent and independent variables
Resultant
Local ring
Directed graph
42:59
Euclidean vector
Group action
Building
Sheaf (mathematics)
Firstorder logic
Food energy
Subset
Group representation
Connected space
Duality (mathematics)
Local ring
Multiplication
Position operator
Social class
Family
Process (computing)
Closed set
Moment (mathematics)
Perturbation theory
Connected space
Position operator
Proof theory
Category of being
Theorem
Resultant
Ocean current
Connectivity (graph theory)
Algebraic structure
Similarity (geometry)
Perturbation theory
Heat transfer
Number
Sequence
Time domain
Funktor
Operator (mathematics)
Reduction of order
Energy level
Boundary value problem
Subtraction
Condition number
Compact space
Shift operator
Sine
Neighbourhood (graph theory)
Algebraic structure
Line (geometry)
Computer animation
Sheaf (mathematics)
Object (grammar)
Family
Limit of a function
Local ring
Inductive reasoning
48:54
Euclidean vector
Building
Group action
Beta function
Orientation (vector space)
Direction (geometry)
Multiplication sign
Sheaf (mathematics)
1 (number)
Parameter (computer programming)
Mereology
Explosion
Cylinder (geometry)
Connected space
Cuboid
Supremum
Position operator
Fiber (mathematics)
Social class
Compact space
Theory of relativity
Spacetime
Product (category theory)
Constraint (mathematics)
Process (computing)
Gamma function
Building
Moment (mathematics)
Staff (military)
Perturbation theory
Functional (mathematics)
Automorphism
Orbit
Position operator
Category of being
Arithmetic mean
Cylinder (geometry)
Order (biology)
Phase transition
Mathematical singularity
Right angle
Uniform space
Resultant
Directed graph
Point (geometry)
Identifiability
Observational study
Connectivity (graph theory)
Perturbation theory
Point cloud
Event horizon
Number
Goodness of fit
Causality
Operator (mathematics)
Energy level
Boundary value problem
Associative property
Subtraction
Units of measurement
Condition number
Addition
Gender
Projective plane
Counting
Algebraic structure
Field extension
Combinatory logic
Iteration
Object (grammar)
Game theory
Inductive reasoning
00:02
after all and in 1 and 2 the most what is sold some in the lecture but we have described how the order by the syringes coterie of stable maps the bond category over "quotation mark streamline operations because across women section front have described the other modulation ships and ports more structures numbers and solved the packaging is essentially reversed obvious so we want to results of oddities questions for hours of the sort of mysteries of covering of pictures that Nelson was current assistance coming from some of the relationship former logs on having some fun dosage which believed in some compatible away this question operator so no so the goal then is soaring so now we have a complete set up by 2 per child per the cat as a subcategory of sort of general Moffat objects into some other celebrated subcategory which actually reduced news season was so that the enough of them to extract extract data from a and B would determine underfunded which have properties the under boxing so that we discussed in length and of course geometrically you want that this His appear in general position as far as the French on the rest of the Freedom Front as far as the 1st serious concerned now if you think of this CEO locally there is a finite number of sections of soda from a number of operations anyone that each of them has In general position but this can be achieved for this it cannot be achieved phenomenal want someone cultivation to the other part of the U cannot in general achieved an position because of all the structures which are but for the Dodgers hundreds of responses we just want to perturbed that that's impossible so now so we have to Petra and of course make chances to do so so if you have 2 different perturbations we want to make sure that that in each connected component of the all space of of all categories of solutions spaces of the compact yeah the Taliban we want to make sure that they can guarantee that they can connect their self as generic as possible to perturbations in the office of compact so so now in order to actually make sure that that is the case you ideally distinguish something like a convex set of provisions that can go wrong for the Reds also clear that up we're acutely with a compact so early them explained here so so we will actually be able to make an infinite construction and then there's some kind of an inductive procedure to 2 reduces the start of this law and things which cannot Split falls off that Madonna contain buildings was hard 1 then you go to the next 1 where you have buildings of maximum height tool but then the the the boundary faces just by the other what structure which you have have to be explainable in terms of the data you already there so you put up this perturbations on the boundary and the extended then you have to then you have to distinguish between connected companies together and and others suspended cylinders and so on so those sudden snow most things which you have To do wonders for on this level so then of course we'll just see this procedure go on and on and on and on so it's not surprising that the perturbation gets larger and larger and larger and larger so so now if I only can control those things in the neighborhood of the year In up priori given neighborhood of the region's solution said you can imagine that at some point to follow but actually happens every other approach so what do you do well you useful melodic knowledgeable so you better to some level no the new court this data than take a smaller but division to begin with can run high up and so on but that would require you already to come up with some packaging ideas to just a and get a complete set of rules of dated so but it turns out that in this in this problems actually over and has but has to do something really it's basically an abstract affected don't have to know much you have to know a little bit you have to exploit a little bit more about sequestering operator when people usually do so this more compact is in the back and actually this infinite procedure can be done so dramatically can run up you never have to start small and go besides unwanted so you can make that Infinite Construction get an infinite source object "quotation mark so the cost could be used other approaches and so this final compactness and could be used other approaches as well so there would never be the need to do what so so in order to deal with this aspect of the story so you have to measure the activation of the aftermath of the size of perturbations and you have to come up with some idea how all that ,comma controls Compact what so so we have heard what what looks to the enormous in the case of an employee falsetto so here it's basically a font of soul it's actually find on on this things which have a little bit better quality on the ball In the fireball but I new but if they're not like the spaces just put plus Infiniti so fond of their restricted to this he is on each 5 unknown but solo if you go to all especially continues to have the foreign policy of properties so if I take a sequence of objects so that any goes to 0 flats and number sequence of numbers and if the debate if the as a morphism class of the base .period below converges then the isomorphism clause of his converges to the isomorphism Class of 0 all the time so it's a requirement some kind of local uniformity if so you have so you take a sequence feels like if there's a finite that means like Trujillo so then this size goes through the offices of sequence of objects the but now you look at the base Of this objects allows a warren of arcade thinks the other morphism classes soon this convergence the other morphism close of this should converge outlook for the love of 0 although so then there's sensitive not going to explain that scored reflexive oxtail all that's important to find actually followed this road to utilize more general competence 1 1 have 1 would have to explain a little bit about that but basically this you can construct always a fly In a bundle category is actually reflects much like him space in our work so so so that it can always be so so there so is a set of uncontrolled blood American pursue talk about an hour about but from the Compaq concessions so then it should be compatible the 1st of all you define if you have an external can define this on the final product to just look at the end of effect of next month's important that you take this synopsis that is what you can control is Mexico's controlled can control and that means that it can control compact pursuant to control compactness of support on the boundary than the front on property owners has allows you to this process control and that to enable because no Okaloosa symphonic dimension times contractions so if you're not too far away you still have to contract of pots and so
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still essentials the mention in the controls so then you want compatibility structures which then you want to have something like this effect supply is the projection coming from disability topped the composition they put the 0 1 form on the Montrose to embark on the trip a continent either 0 or the other way around their mother was 1 of all structures things you want to have this it measures the on the other nontrivial offers 1 form on a nontrivial unwanted companies all or in this case over the cylindrical part and want to behave like this the reason is that he later Petrov anywhere there's nothing on that side because over the selection the perturbation of 0 would you like to do all this yeah this cat Oh yes fourstrong strong bond carrier wall has but by filtration evocation of sleepers and plus 1 and if you have a strong bond of structure descendants he always it also gets a bye attractions so on understated Mexican talk about the regularity of the object and then in the fight than you can in there and if you have something in the in the in the country far far away you can talk about the regularity of the 0 1 form all the underlying object and that has but in addition to his who allowed the action the young because this is only it's precisely finally we so because of fog there agreement was lot yeah so the memo that as cousin encountering pondered this or that for strong buying the that this and it's not just about but there is an additional this book by filtration and if you lie on the 1 level itself Compaq perturbations in supply sections of all sections of the 111 so that is a there you know that things are compact so you soak in the final year of year everyone 11 and so on but if you on the 1 level you go down by Compaq is on the 0 level and to put by sections 1 and that is precisely the sections of the mother section but we were given by sections of this part of the images of the section into line the better part of suspects that so that is the holder of the EU ,comma contributions pollutant and then In addition should have this property is so if you forget about the stuff over truth cylinders and fix the norm or you move all you even saw the continent away which of course is what has been done in the trees in the continent and then you see the same measure so then of course you want if you have but if you have a disconnect the conflict true configuration moves in against each other then you would see this measurements Of those guys the next take a look I think the only I personally was so no 1 so so if this 1 that out and on and on and on and so this elementary components so they and then you take this and then there would be some cylinders fugitives :colon some Salinas which you use your could spark in order to compute this and if you look whatever those lunatic that's so that they say earlier that and what have you as a nontrivial so it is not known to have so once compatible reflexive oxygen on exists as a construction is rather easy take a lot of models and then you you will locally never retracted abundant space just takes a moment petition of Unit 2 times in all as the new restricted to military nuclear ended up as the source of this problem so it's if you have to construct such saying for born of many fault Bonner bonhomie was precisely what you would think that might be a little used to be set of just so yeah that is what I mean they have a constructive solution although constructed like this and it would have the property of the quote from a general than if I do the same procedure of reflects a fight about this procedure prosodic produce the norms which oversees the property which I want so but but it turns out that extended the flexible work have some additional futures which I have described to small properties can't because many of reflexive fibers can start talking about the convergence the 1 and that is actually was useful thing because the opposition Freedom operators herself close grass and then you can even if you don't have a true convergence to conceded that in some big sentences Woods satisfies the creation means to get better organization of compact sport but survivors of some time to explain so once they are subject to the north then they can measure the size of remote section from transform also 1st of all instances of founder of a composite of orbits spaces In the negative didn't function on the obvious space of stable maps just by looking at the maximum of an absolute value well I look at all you will which is Nunzio so basically I missiles and at the maximum normal to different branches which are milk the timeliness you're out here says the plaster meant that but this is a section found about the local picture is that is a Cplus section so they go into the 0 1 5 so so focus on Arkansas but his attorney has a using insulin resistance what that was the 1st time to think about yes but nevertheless central government managers say that here so this is supportive for bonds from 1 was so then you had this filtration their cases in France and the last 1 so so that source of equestrian operator rose from the endeavor to the end to the end and but there is 1 bit of fiber and 1 so you could you could look at sections which here and this also compact activations because of the better regularity and you can go down to the end and 11 and then you get compact In the plus modest sections locally are given by sections of the country so there really so far they have sections and their lab precisely in this no 1 takes real values on this side of the canal and have this picture here at Oval in the southern waters section structure just look at the maximum enormously so the function of 0 1 since other objects have the same old it just as a function telling over what was normal the next enormously over a dozen of his what at it while you look like you're wrong so far so given so the effects of such a long and now we also have to exhibit the compatible neighborhood you will recall smaller space of their growth so that they can guarantee a certain competence properties for that part of the problem so so I want to say so so but I want to say for example this is a neighborhood of this model space and you have a perturbation which is
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supported in this neighborhood and along the northern measurement is smaller than the solution spaces automatically contacts possible Of course if the neighborhood is too stupid it is not compatible with all the operations have like if I construct something here and it is in the neighborhood that bring it to the boundary and then this boundary part doesn't belong to the neighborhood of the next level than I'm stuck so the neighborhood have of neighbor of the solutions based say on the level at which never degenerates then it gives a product on the boundary of the next but that should be in some sense a trace of the neighborhood which I have not so that so this was did you know that and they a lot of activity along with 1 of the young so this is just 1 of those origins will Mulford object to is the lowest in the world right now we were so so this is the sub space of all of that that clause subsidies of this region each connected component is compact the plan yes 2 so now I want so so now compatibility so I have to so the compatibility is best explained by the hierarchy of was for inductive constructions so this is a set of self connected components of disease they then this spirits introduced of those which never frozen desserts only unified won their own they contain building of maximum height tool but through the contain components that Mexico is tool but there is a building which of 2 of Part 2 on the top floor number 1 and so on so that is also home to reductions goal here come objects situation can count and then start some schemes so this without some intermediate steps I think would be of interest to soothe something this that level work so so just so this is sort of the mother of all schemes but nothing for I fall took off pole for certain specific seasons have to subcategories you might be able to get more structural of but you have to adapt to perturbation so too is is not written in stone but that is that always works but if you want to have more fine structure you should on certain subcategories modified so there but then we know that have been we have this forgetful font of its forgets unstable the river forgets to Roslindale of buildings if you pass a slow and this actually MEPs connected components to connected components actually preserving this the composition so long ago from a company in which I have so and some traders selling also just forget the truths us and then the image of another company and then he writes in a few words of is a subset of that that representatives of you all principles limited run like this and it's a basement here so so so none of the stuff I mean the connector companies ship PSC indices and somewhere storm so it open subsidies competitive provider so if it after forgetting to the confidence again this club then the open said over a prime Of the cranking should be just the privilege of that so that means the neighborhood of attributes and those not not restricted at all but I don't have to because I'm not going to patch up the true and that the ball was a 0 0 then I always find my so can be generous on that side if the objects in a have more than 100 companyowned and UAE's environment as local family structure and I moved things around against each other and if a so for any aid it holds potential for the for the faces of this aid holds that if I look at the category this objects in you would that fiber products in the country new take take orbit space and put this back by Of course the image of this isn't a subspecies of and put back by the covering functor Ltd which basically means if if I have been open said he opposite here takes a fiber products and viewed as as a boundary phase then the neighborhood which added the crew complement restricted to those responded the source relief construct the lifted up from below not that clear this inductive thing so you start on 11 0 so if Oprah to open neighborhoods of the city of 4 million new model aspects then you go to the next level there's a face another face pulls back this part of the 2 neighborhoods so on the on the boundary of a product of 2 neighborhoods and then this should be the restriction of the neighborhood are that company oracle "quotation mark himself given giving our reflexive hoaxers so this is important but you can ignore the will of the people of this state of the state that it would be wrong result that brought us some wanted snakes out of state if it's a Southern thing you wouldn't realize and I could tell the Americans would have told you wouldn't have figured out but that still but I wanted Syria and so on so on on the level of lies which are hard and scores noticing that so I could have just the same not mention the mind no I was referring to being in the ad but they have not been nominated another user licenses state to get the point across but In oddson approximations of insulin so I I I tried I didn't want to lose an approximation the so then in northern subset so if you have an open subset of the orbit space defined so the extraordinary numbers and define you into consists of all isomorphism class of objects which are belong to you where the Norman equestrian and it 2 1 and the Triborough Salinas occurring at any fraudulent anamorphic this allergy Abramoff examiners as much stuff here and this saying here over the years has some data on it so that it applies a question the mine operator and and Nevada its knowledge that's equal to 1 so now the base is that this is is accused developing an open subset of other classes of objects so now associated is open said you just take all objects in this opens full its the the truth is selling those every flood your home office that means if you applies at the box there's no contribution of attributes the new apply it should be less than equal to 1 the the head of the the Federation of American Societies with no conditions no 1 knew that I had concepts and none of you was no I give you open said I don't see you when the result you and is not openers is Devon because of the condition of any and have loaded that would less than not than I would have to allow basically every student at on neighborhood of those coming out just because so seoul the thing is that the perturbations on molded precisely producer since then narrow but this time over the last 2 years so this the way when you have an
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action that would reduce the political and military now because if if I have a building supports the sweep floors and then have assumed that you and I do the growing construction I need variation of those guys he always says no growing if so the thing about this picture so and then maybe something untrue here than something untrue but he has attributed to the and then something happens here but I do this part he obviously in the monarch grew this thing has to do it did close to this 1 starts of the deviating Sony functions if you do this and in this the set under this and I need to vary the function you know it's prudent all this is knowledge so I need this various things in between Indian and the functions space in the broad market in order that I should forget federal 12 was there was no you don't have to watch enough import In the middle of view just below that is the building 0 0 and the New year but it's not only the plant because of the quality of the an yes yes of course as a Muslim so I don't remember what it was like being well just as implicit in the center no I'm not not symbol that just Minuteman I have so I always distinguish between traders cylinder components and the geomorphic summoned the trees and the components of top and bottom does the same thing from topic to 2 a general of soon this is a condition that has beginning he said the plan that you lost the game yesterday not here so silly as a foreign stock rose from compactness property giving the effects of compatible Suriname on and there exists an openly about you because smaller spaces the unperturbed German because monasteries such that the closure of this you and which are those described as a foreign property for every connected component the intersection is compact so so you can choose neighborhoods besides it is a properties which are described to rich that you live on inductive process and you have compact going to compartments it's a closure of Singer said influences that look at all of those things the because of this associated gone so that ended can choose to be a neighborhood which is compatible which is compatible so enchanted this
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conditions so so that tenants another 10 that has you that new inductive arguments the neighborhood still working and then appear as a product of the restrictions of neighborhoods who had already there and then he he extended ended but to Beijing by the by the construction before the UN is the product of the perturbation by the coming transformation because of the definition of the maximum of 2 twisting has same along the boundary so it's less than 1 so I suppose the 1st attack a sequence of epsilon allies less than 1 on the lot so studies if some 0 you patch of smaller than epsilon 0 then you get something on the boundaries more than absence 0 then extends smaller than Epsilon 1 and the next assumptions wannabes 1 extends more than or is controlled by this neighborhood compactness never but it but I explained that In order to so so then there 1 issue so that the 1st said if I have such a neighborhood you which has property estate and you as in this hearing that a new controls compact outsourcing that's a good word of controls compactness all of the situation in the city last year Armed yeah because of reflexive Foxrun it's almost true but there are certain there's a certain problem the transition transition to the new lift opposite covering and want extended to the bombers some it's it's it's a spilled is a the office supplies it's not true and so I I I I wouldn't bet my life on it neither you along the that there could be than such a property was followed trucks and announced good response who's on the air I actually know what it's like in this room alternative would have been nice to have but I could not himself so now here him here as 1 additional on which should part of a new venues forgettable itself if I have the perturbation below and it's it's supported and you will hear incident his support and some new prime and I takes a product go the boundary this is not supported in you cross your prime if you take a product of 2 compactly supported perturbations of product is generally not compacted support the solution said is still the same because you could be the complex support and outside its anointed system intrudes support From the beginning said you so so if you have the line and this is a support cross the line the supporters here then the plane plaintive looks rather like that but support but the solutions exist worsened said of course for the products still how so much so that in the past 2 months and fined him so its supporters students the decisions of the evening of supporters here in Europe before the promise of supporters and dismissed from their plans for a while now no no it's a functional form so to have a function from onto our cross a function from onto to also get you get something from our into or the article to the problem of this review is that I have that none of the but you have a pet yet you have a friend who have a front on problem you have is a compact is supported by the Britishbuilt stateappointed that the products into the product for the from would not become competent support I everybody likes this fact so you find a lot of people's blood but it's there I like flags up not just said contract support absconders companies companies reports of so everybody this is generally accepted that but I think but you have to believe the care of the he said that solution set it up was a product of the 2 solutions to it a perturbation was compacted support and forces you to have a not so if you know that for each of the firm problem the solutions contract and the Protestant airline you want to know that means although what is supposed yeah the net from the relentless reliance and the supporters here the pose of another member of the supporters who takes account using part of the 2 things get that it is means every F of X ,comma GUY so it's man from ought to withdraw to to review right so then then the supporters here and there and that is precisely what we have here so however you can use the official said In the end of year you know this is the a future the world the Phoenicians who landed in saying that I had when you were a lot and I think this
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will be over you don't is not yet UN could survive officer classes in you so that is true and going to sing on U.S. troops in the middle of the day said that a little sense of
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London yeah it's just that this is an excellent nonusers for section so the like section found that that goes down to all of its space and you just look at all the object is to get the maximum normally 1 of the branches and that's invariant under isomorphism soared To get a function on the altered states so far so now however if if you are in this situation and you know but you can bet that this space so if Islamic give this temperature so if you will so so suppose different problem this and then you take a perturbation and you know that for this perturbations solution said that you allow any any 0 and 1 is Compaq so if you can switch it off then if you didn't have the there's the stronger assumptions Osasuna this is competent but of course it's true this kind of always benefit it takes a product which which would not have Of the perturbation take to subsidies suppose as of 2 years bonded support as of this of excess bomb support then of course this was happening here but it would be true this told Parliament as he S but because it can hold trouble you can put them there but you can't put a bomb function for which actually supported the deserves identically 1 heel and to cover everything off around it would not influence the solutions while of that is even unimportant so what is important is that it means if you put so if your players so now the plays if he takes a product take care of it just put them in front of this perturbation the couple functions and offices about a 2nd problem was times as prime the studios and comes along so because of that property under control compactness you can if you take the the set of solutions of this problem for minors onetoone then you can actually cut a fine ,comma function just under solutions that has to be 1 can cut off so so there is not a desert different cultivation which actually would have bonded support it would have the same solution said and the same unionization and solutions so so there is this marginal difference however long if I would require this then this nice multiplicative property goes away so I if my perturbations unmonitored definition but for the fire on the phone support Arkansas and that it's size of 1 which has the support but but if it has the same it has the same function than a part of this and other perturbation inside if which would have supply so so therefore that isn't the point that has to to observed so without loss of generality you may assume that as far compact is is concerned the supporters in and then we have compartments so but so so you need this property you know then that this is where the hair of his soul so but that precisely we have because I have estimates for my I have a statement about you and which precisely says if this year's positively it was precisely says this
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of citizens some Oh here this properties but this means if you putt this by teatime says set some section which has lot known as 140 being 0 minus 1 of the property so it's better but it so so of course I could have gotten away results saying this because nobody really cares about that but uh but itself but with little so but for the following arguments I always say the supporters you because I could modify so far the
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fungus have compatible accelerating a compatible neighborhood of this new controls compactness now this since local laws of various intimate spaces if he was petitions of unity there so that the support an open final of course at this stage are can be used to patch together and constructed this applies not section find also explained to last summer to construct against constructing controlling derivatives and whatever everyone so precise it to whatever you want to do and find at an inch so the whole whatever you can dream up and dimension can responses on interleukin2 on this level so because you have the same amount of freedom as far as part of provisions so that take is life so what is the last In the end sometime year so it is likely that every port at every other month for the glass a less 1 the In the SoHo None None men if if this Is the lesson 1 everywhere I ever ever ever had and all of us and but and in addition it has this problem which would be precise been example given just takes a product that would be a perturbation that she what nontrivial In the end rather March said when it would have problem the technology Poland has signed a none of the euro but that's precisely this problem for each branch so this is also a scaling of the branches to 2 in front of the is 2 times the is a rescaling of smokers sections by multiplying the line has a lot of things model No this is still addressing the starting point of my discussion so far so if you have this property here then even if you arrive in this bad situation you can find a new perturbation which has precisely the same solutions that was insanely reservations which has which has supported you so so so sometimes crying record source I'm not forget about this fact in mind active procedure and overseas supporters knew which literally is not true but would have to be true as competitive as other structure then it would but but you can but since yourself can cut it all and once you a little bit inside saying the new Kansas in gold yeah so to have this big support here than you go into position on the boundary than you go inside just merely decrees it was something that you understand you "quotation mark so not only almost sentence like final dimensions in the case of sections so obvious
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perturbation frozen some the following the results given the compatible UN so the season constructed they Texas for every absolutely no 1 the compatibility of supplies might section front of competitors overseas what costs here control the and you satisfying moments as an epsilon so that London and the Bonjean position that's the basic resort to you here comes another proof so let's recall this then let me define 5 your Leslie Kosgei just union of All Saints which have to the consists of connector companies but the next current floor numbers change so that is also holding 2 reduction then there then of course I loaded or I don't know why I wrote this forgettable so I could so we say a has at least 100 components if there exists in all of for this the African 0 Onava into a nontrivial compliments of the enormous orange this say tools classes are related if the image in PY 0 it's the same but this thing and this saying just this difference by shedding a certain number of Chavez and abandoned then there are the good jails just the restricted to this and this thing just stands for the Union of all so so as it's S is the 1st subcategory associated to or objects which lie in some of it in some companies which regionalism another Mofaz applause lines and complement London JCS applause among modern section from defined on this and been inducted procedure starts constructs along the 0 then I go to the next step it extends what I had before it goes to the next level and so on but inductive statement yesterday increasing secretive some James 0 only exists for every Janice about legislature from him the foreign properties 1 level it's restricted to the previous it's the 1 which constructed the on off the sing up to leverage is less because Jr the poor the boundary of the singer leverage supported Beck of the date of which constructed on level up to 11 Germans 1 of because Bondra faces coming from stuff below then this restricted to the gesture betrayal then this has to be true which precisely says that I don't perturbed over use of similar problems and is compatible with a local family structure for some reason it should be you would be this controlled by a new allweather so landed the Compaq his control over this said over the clout the close of a subset of the the orbit space associated to the connected components in and this sings on general position over so if you haven't got it if you can go to the next step than ever perturbation which produces compactness and but then a lot of statements that would basically says but it's only the let's start so the induction 1st of his induction
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notice that isn't so take 1st the company is our tribute stick faster connector company which does not represent object mistress compliments so this other things which can up compose father I have an open neighborhood around the solution said you which lies in Of course I can have only 1 connected component the Kosovo would have to win mantra could shift the gains each other get Bondra strong I mean the shifts and so that get level the together building was to floors that would have Bonner's starters like that so but actually if I exclude travels the Commons at only a 1 connected component then you'll find a small perturbation on that understand this known lesson epsilon 0 so that of this he always there a general position and the and the support of all the sections occurring there is it's you but that enabled the job construct now so this is this is a basic statement if you have a friend an operator of this contact solutions said to take it the neighborhood of this city takers perturbation is supported his neighbor or the new vigor little and sings becomes principles such as the classical unresolved just like that so there the notetaker component that is related to a so this differ from this by having company so this is how we define you just takes a perturbation which you had on end and on the q ended the extended sold at all attributes Linda there's nothing OK so so now this is obviously translucent as well because of the perturbations stance Rosalind Salinas because automatic transfer of energy and otherwise it's the previous day so then the support of this lies in you because you were in the sector is a prime because there was no constrained whatsoever with respect to the city but compatibility conditions so so now the connection of these defines London 0 1 and it satisfies 2 0 0 4 but straightforward so I'm not going to be a
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positive association now can a lap it so it says that in a consistent way I can put on each connected component like so there isn't as simple as much as 6 confirmed globally which cut tops the Fratton problem general position that it has the additional compatibility conditions like the data before bid when occurs as a fiber products of another company and then the perturbation there's a pullback of the staff which was constructed then if I have the ability to the chest when untreated combinations and moving 2 games each other and that was a solution that was a solutions also solutions but if I had if I have a solution traders and the judging from all sources also so this is precisely if you have this in no way porous so if you have this thing and to comply this Swiss cautioning operator you get its name was brought against was weighted sup the categories which locally accounts can be represented by this so far many of besides Woods and this other sings it can integrate always careful traditions and so on so that is precisely where you can solve do what is in most papers with nobody we could do with counselors oddity and ended up at this point and they say what they would do if things would have been generic so so that there's a lot of documentation about what you have to do so so you have this can basically actually take this paper award as if tea and just starts implementing the but then the clear but it's so now so soon this is constructed resists the cloud properties of enormous for example as a so not only saw no take a company and a 111 Ohio forbids the objects are connected so which means that if you if you look at it had been dignified 1 it looks like this of course on the boundary might have to composers but it's still connected to the bone so now consider the poor back via the with other covering fund does all the previous cultivation falsity down the face of a so no for each source so the seat that debts so you soandso phase comes from the point of comes from the pullback of all of them 2 problems which I general position and then suffered together to work toward a 211 1 buildings the 2 of hardwon and general position on the boundary and then you have still the normal direction to the boundary into Petrova to move general positions and enabled the suspect from pictured on even so near the boundary wouldn't even need this because already onto the and then extend so so they so if you go mindful of what 1 thing when all of the ones should point out also that if if you are so that things 1 should think about it but they did they not because difficulties so so I started In the induction rivers the eh companies which don't decompose further and obviously I can achieve gender position then I take a product of tools such guys I I'm on on boundaries so we have a part of crossings with the news agency subject event if I just and put it back then on the boundary it viewed as an operator with respect to the boundaries already and then you can extend its someone nearby with dissociative and then inside you have to figure out a little bit subject the then so no 1 happens so what happens if only corner so I'm calling can argue like this then this party is restricted to the subject of and restricted to selective but if you live in your eyes with respect to this direction this direction or precisely what the liberalization you cannot say no my 11 additional promotor 202 million subject to box but so you have to live with the legalization of the and this automatically so to so so then the restrictions of previous things are subject to certain general positions citizens find new and extended "quotation mark also among some of the world's but it's so bad this I'm astonished was shot approved struck with could put itself so so that the days of faces and put back perturbations of so this was the 1st result Tracinda components but that From silence he extended the 2 related there's before related means I edge chosen to not take the company and also on the same level as outrageous abilities but disconnected conference then studies of various services sector pullback back of perturbations people will laugh and it turns out that this same here by just because it can decompose into disconnected components which I do and just takes a product that relation was already too on the bone this just extended so then 4 then you have to brave about his words about some of that Wilson was just went on with millions the and sold them that's defined and then but then this buildings here I extends by adding truce industry so now so now hear that take products and so on you you sold but this is .period was when I discussed here this difficult which miscast here but if you believe me the acquis of modified and bring it it was something the which has supported you haven't than this procedure never they always have 2 sections which have supported you and small an upbeat over the compactness results says it must solution spaces compact so then this is a step and is an extension of the previous 1 and so on so many of transceiver so now remarks so I started 20 minutes late because of on the a 6 but got nothing but so sold so I had that so I had actually 2 endings for my talk and 1 inning was to show you the home atop the argument that election between 2 perturbations more interesting than that once that's Wallstreet for sold you cannot you cannot have general position in your home atop from 1 to the other and the reason is that you have 20 dependent so if you if you take if you go for so on the 0 level you can the holiday can hold of nicely but you also want a home atop for example like this but if you look at the projection that you always see more singularities for In order to look at the moment space was removed when most similar so then you do the products and go to the to the next level but then of course the perturbations of fiber port was suspected to that cause some constraints in general so that things are not in that prevents you from putting things into general position because of all the other structures which and the problem of precisely situations like this there you so also if you have 2 different components below then then of course this is a minimalist space enough to you can talk about that this is only see most of singularities and on different continents which are really different you can make sure that the critical levels of different the generosity but but he had a compliment and the address of industry is the most functions the same so this seems already produce the same critical port critical parts of it like this suppose to have this thing said beta users of political gamma so then you can't do that say on it and you have a veto if you're going to put a ceiling on this that this thing classes cylinder here is related to the object this differ from this object only by adding attributes and then doing it so the perturbation you see here he has the same perturbation you see here and a bit of fiber product that causes in difficult but the deficiency in the Corcoran is 1 a normal direction if drove to sings can but you have to find out what a would perturbations because many reiterate this process the emphasizes this kicks in this from so so there was a lot of this is that you can achieve general position but you can achieve identifiable garbage class good position and you only have to integrate the boards good stuff the others just so and identify regarded as something of a you might have a solution he said but where there with the solution spaces being would be something like that let's also the stock assistance onedimensional because all the other stuff looks like you know you can so the garbage out and thus I mean if you just interested in 0 automation components what happens is precisely because you you might have this picture that technically it looks like there would be something like lots of space so that doesn't play a role but then when all you have something like this this is good but of course in general position would have been something like that to come out of a corner what is on so so they have to take action explained on the solitude during moment of it this was 1 inning but after my disasters for lecture but only came through Harvard's 100 counselors and deserving of 2 counts of plunges here well because all chairman's with so the last on so so he is so that the concerned just make some prominent where would have to prepare for the discussions chose this analysts say so and at that point and you wanted to do was in hands so what you have to do is but orientation things in Iran receivables on issues and the best thing is to go to covering the Indians of TPA pavilion number the ponchos the predominant imported markers you put me random marked points on the periodic orbits some requirement then you see that debate is occasionally ended except Brazil crumbs of bread orbits and then you can ride on Inter and then you get the relations but more spaces for their translated and then you can represent the data in whatever said then that's the presence yes no 1 has the right did the same thing will happen with constant cylinders so I will look at the end when I was a like the Our 1 I'm old and new ones and then new unit you definitely have to put so so when I look at stable maps then I would have members from some of the highest in what is happening in the world for the 1st it doesn't never crossed by itself on cross in combination with something else where there is no way to know but but but but the construction you have is not a product construction so in the future you cannot you cannot have the Trevisan and Abide said because it's an unstable object so so attributed some money but Treasury but cast all of us together with an untrue component and then there's been a constructive global it's not a product of a construction is Selinda and the other 1 it's a long it takes bolstered it takes data from boson mixes them up it's not a problem so so therefore Jerusalem Salinas never appeared alone because on stage they can only occur it appeared together with the Montreal object and then when I makes a slice suspect because the Lakers lies with respect to the on iTunes that you would you fix a Vermont report but you can slide the cylinder against yes yes and that has to be built in but it's not a product situation because you also have to divide you have to make do you have to take it controls constrained you have to 1 the automorphisms room before this land as well so so there's a certain things which have to do at the same time which going to the construction of all the uniform so severely hit by the blast and you will hold which is the with really