1/4 Polyfolds and the construction of Symplectic Field Theory

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1/4 Polyfolds and the construction of Symplectic Field Theory
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Hofer, Helmut
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Institut des Hautes Études Scientifiques (IHÉS)
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Topics: Short overview. The category of stable maps. I/O structures. Covering structures. Additional structural constraints.
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Logical constant Axiom of choice Polymorphism (materials science) Complex (psychology) Group action Mass flow rate Correspondence (mathematics) Thermal expansion Sign (mathematics) Rectifier Object (grammar) Forest Negative number Bijection Physical system Social class Spacetime Sequence Automorphism Orbit Category of being Cylinder (geometry) Order (biology) Abelian category Stability theory Directed graph Point (geometry) Connectivity (graph theory) Translation (relic) Number Tangent space Frequency Morphismus Latent heat Energy level Selectivity (electronic) Lie group Units of measurement Standard deviation Matching (graph theory) Prisoner's dilemma Surface Physical law Algebraic structure Line (geometry) Mortality rate Equivalence relation Local Group Sign (mathematics) Field extension Summation Equivalence relation Social class Subtraction Principal ideal Rotation Building State of matter Multiplication sign 1 (number) Mereology Positional notation Meeting/Interview Homotopie Arrow of time Position operator Stability theory Principal bundle Logical constant Process (computing) Moment (mathematics) Perturbation theory Functional (mathematics) Connected space Maxima and minima Well-formed formula Isomorphism Condition number Energy level Trail Vapor barrier Real number Modulform Heat transfer Operator (mathematics) Isomorphism Subtraction Homomorphismus Condition number Shift operator Projective plane Number Positional notation Computer animation Commutator Doubling the cube Morphismus Bijection Negative number Object (grammar) Fiber bundle Boundary value problem Spectrum (functional analysis)
Axiom of choice Rotation Building Group action State of matter Direction (geometry) Multiplication sign 1 (number) Mereology Weight Asymptote Leak Explosion Positional notation Object (grammar) Number theory Length Category of being Partition (number theory) Position operator Fiber (mathematics) Social class Physical system Logical constant Spacetime Product (category theory) Complex (psychology) Perturbation theory Functional (mathematics) Bilinear form Automorphism Category of being Isomorphism Funktor Order (biology) Chain Normal (geometry) Condition number Abelian category Identical particles Resultant Stability theory Length Directed graph Point (geometry) Differential form Trail Modulform Algebraic structure Number Frequency Morphismus Funktor Lecture/Conference Helmholtz decomposition Isomorphism Boundary value problem Energy level Contrast (vision) Subtraction Associative property Field (mathematics) Condition number Series (mathematics) Matching (graph theory) Prisoner's dilemma Physical law Element (mathematics) Algebraic structure Weight Line (geometry) Mathematical model Affine space Field extension Anamorphosis Number Positional notation Computer animation Morphismus Universe (mathematics) Element (mathematics) Social class Finite set Object (grammar) Boundary value problem Diagram Matching (graph theory)
Point (geometry) Building Vapor barrier Multiplication sign Function (mathematics) Mereology Rule of inference Number Morphismus Frequency Funktor Meeting/Interview Object (grammar) Devolution (biology) Bijection Length Subtraction Fiber (mathematics) Condition number Physical system Product (category theory) Prisoner's dilemma Physical law Projective plane Element (mathematics) Algebraic structure Price index Perturbation theory Bilinear form Automorphism Category of being Formal power series Computer animation Isomorphism Normal (geometry) Condition number Right angle Object (grammar) Identical particles Directed graph
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Complex (psychology) Building Group action Greatest element Multiplication sign Direction (geometry) Modal logic Sheaf (mathematics) Hierarchy 1 (number) Insertion loss Food energy Traverse (surveying) Group representation Connected space Cylinder (geometry) Descriptive statistics Fiber (mathematics) Stability theory Social class Process (computing) Building Perturbation theory Connected space Flow separation Category of being Arithmetic mean Cylinder (geometry) Lattice (order) Vector space Order (biology) Linearization Right angle Resultant Point (geometry) Ocean current Identifiability Connectivity (graph theory) Distance Number Frequency Energy level Associative property Series (mathematics) Statistical hypothesis testing Algebraic structure Division (mathematics) Mortality rate Line (geometry) Summation Equivalence relation Computer animation Object (grammar) Family
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is so great so I think that in 20 thousand I was convinced that I could do as if to solvent something pages for Adidas but but right from the beginning of sort of feeling would resolve the wrong message to Lewis and if you have to write a lot of things and you never convinced about the project was really also so sobering and then this results consumer the underlies all of what are the issues in the safety and as if to say it is it is a really nice problem form basically every aspect you can think about it namely that it it has a lot of analytical difficulties but all these wonderful eligible structure comes in fact form all the difficulties you see so so you can't say that put money can but you don't get worried father put additional proselytism and some of the difficulties underwriters and then they can do a little bit and somebody wanted during full strength you you have to deal with all these problems and the problems in the under the constructors is that it summed sometimes quite 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technical issues wouldn't have arisen postings would have progressed different so and now Howard Hughes is a classical from some form if you it's such a classical those qualities was not difficult status more perturbation of the section some smelters awareness of properly that emit commits instrument soul but perhaps looked at this result may be no 1 can achieve this as well so but what are the problems of the problems were course that's corner changes on all the differential which of course is the reason why I find that the introductions were introduced in the 1st place because you reduced to spaces and restricted disposition obsessions of canonical Oslo's OK so so then at some point in this project became so once we on some kind of a set what was talked about last week that it can generalize differential geometry of the local malls or redirect then then this part of the project Summer of Goliath on its own so we really did differences a tree but we also note that all these cases 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this would be a different object if I have an automorphism of this 1 that requires that this matching is being preserved so for example if 1 cylinder is simply call but the break is not a common the top is not cup then this object only has osetra so even if he is simple ,comma the double ,comma periodic but in double coverage to lean on top this has only said it trivializes properly because I have to preserve so I can't really wanted downstairs but it's isomorphic to another object which has a under hearings soaked in this business it will be extremely important not to confuse being the same and the Asimov because for the counting things because when you do in the fine points of perturbation theory if you will the sloppy about it to start counting Bronson but itself so this is not a category and clearly every month visitors isomorphism because it can ride on the inverse immediately by adjusting the biologist Rosen's yourself of all of our pride and voting the body more vignettes now 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that would be the number on the top floor so look invitation go then that they having this grading my kedgeree decomposes and any artificial find what I mean by the boundary of the category this all this objects in inside find 1 so then there is also the important points there is also an important point namely if you look if you look at the all space all that class of this category so the underclass consists of all clearance classes of isomorphic objects then this is a sector no I can the remains of his life in Austria was sentenced to realize a monstrous was the scariest sect yeah the if this is the man in the last it is the I'm talking about is not your soul in contrast has since the beginning of the 2 would be in order to pay Bolivia of the the USM traffic since we have not only number but they have markers and affixed to the points on the parents who don't that at city that we had before as well but he had on the top of the hour mark onslaught of the punches normal and the role of the asymptotic Martinsons upon shows Bigelow's pursuant to hit a particular period port .period on the political front is an you don't need to know what the question was he he obviously read the book this is not so when you ask me I the way higher market this said the news yesterday just made of its sponsor toward and then to be hat is a lift of this it just says matches blinds up to rotation and then if you too if you take over the multiple records for the negative 1 adapted to this thing if you run along 0 0 in the direction of plasma plasmas affinity than in the same period all at the same point on the border what is more than what you can do with the rest of we had to be head has to be preserved so assault on the list of all of this has been a little to early the so you if you have 2 of the slain which generate this class then if you apply this morphism he'll find my mum's 1 idea that it never is mapped to a pair of lines riches that the soldiers to the object it has been on the release of the the deal was I are you with 2 . 2 then 1 match the statement the White on the Net different matching the full of land that I know of no I will just say that if you if you for example take so some positive Dr. Collins economists and then he or something we're covering numbers 1 it said yes so far ,comma this 1 so then I suppose I have just mentioned here then obviously have also matching grant this rare the 1 appears to have the murder of cultural but then you would you what you could so someone matching is this this notation the other would be put this thing on the other side this consultation listen thinks I'm actually there is no maps the thing that's not something world but that that the not the something is some stuff and this has already given a match so the law parts MEPs should if you fix storylines here so things should never to align and the other to align their which precisely gives a miniature in it's a different object because all the data so the person is not too difficult so if you want to question that I think everything in the amorphous thinks everything in the face of I think that the that the United States Britain and I need to accelerate the European 1 think of the climate of the but that it in be so so the 1st thing is N and given object things would be fixed so now if want to talk about what within this you have to resolve this state now would you ask maybe that is what Fort my father's is fixed and I want to build nozzle objects and of course then in fact what you can do is you can take a different more fizzle on each level nepotism singers and adjust the markers or that this is isomorphic was installed last year was seen together with the choice of sausage and time to make a gesture from the university using them for capsule it's just that in the just now and I don't want intruders something which is discover :colon he also so this would happen automorphisms and this it said as well but as the same in this market the optimal 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hold few consist of they taking a partition of unity for the 1 and moving north nor its sponsor presented by several major fractionally life and moving it into a better position that its use of the things look like the metaphor so that is of course some technology to do but this I II
III French folk have not come from you said that there was there be a the aim is desired said before it showed up here so if this into the seat of his additional structure from the chicken derisively diminutive construct of today so it turns out that you can you can points because can define differential forms on this categories it can integrate them so there a really cool and series of so what clear so now so that more structure here so then I have an object in my category led me to find a different objects of different category PDO which individuals is all of the following the consists of maps from finite sets into patriotic and the MOF prisons or by objections will be in the final set so that that is true so this is actually the real already part some sense of the data name if I look at the positive punches and just take this points you as much .period it's nice at and the evaluation plus this this here which associates to each other's points a punch In the same year the narrative not this category might for example if if the different point in this for a finite set go to the same period always you might have different morphisms example if if I have to .period and they go to the same already and switching between points would In this title so in this category I will explain everything to you and then I would say that the notion of a small structure on this category and then begin often explain a little bit of a book the dancing was versions of this like not like from then 117 structure cannot talk about differential forms in this category for example the put back up by moderation or from the from many fought over the French a form that is different from the continent and many of tensions a integrated did I never have to leave the assault once you have a point source it later called the upon a foot switch on the category never have to leave or consumers other construction to convey to the Hortense was ultimately the same thing in the world there nothing new in active use the well overnight I than I did make might've initially assumed that it was an unusual form of most modern and that residents don't want the things also your so that as was discussed last week so so we have on the automation and for the state of Himachal instruction that defines of so-called derivative feud and that has prevailed Roberts but so soon now so now sorcery have an input and output of a just look at the restriction to the punches because there's no real reason which says to which political and then became the big chains like this at this point the fiber products and that look like this so so observer feel that such an RBI which introduced before his BIA lies below the knows of baguette about the mention of the lines is in fact His effect such a so so we look at of the look at chains of objects and his officers academy will be that is called and it takes a final report no result so we have an object the method forwards which would give the net which associates to the problems the participatory role but then we have a morphism from there to another such as zinc which comes from the next 1 but going down so so we have here that different this enough for us to be heavier different buildings object sources objects in my previous category I met them down the into this category here and there is the map nothing to something which came from the negative declaration so note that this singular it looks like in new element in your category but it isn't because the head is missing here we don't know all we don't have information how all the lines of the match that is not there that action will lead to the fact that that there is a relationship in all previous category and this but the US some categories will be of some covering front isn't clear None of this is not a lot of the this time around I mean this is discredit comes from the association has said that Europe had had no you don't because if you know the boundary you have but then I have to related to the 4 categories in which it says is if you have a building in the boundary then you have to related to S and fact pocketed with this valued shop you thing about 100 Chopard apart here but by month to do with this what does that mean you just took the lead for you don't lose a matching condition yeah because In this case at the end a deposit near the fungicide don't have I didn't keep track of now in the original version the head of Enchanted markets so that information was lost but then you get a car in the other direction because because then you have everything market and interviewed together you have to forget together at the on the periphery of the new that you get covering their so so there's no free lunch but this Singh that take all the known all the numbers away all the orders the way has the advantage that I have a lot of other more monomorphisms and my perturbations have to be compatible users don't have to be competitive in the other 1 that but that's a little bit hidden because of all these additional structure than the apostate if I take you all the numbering away from this object only here that should have been part of the same way why not just for the tonsils was out to just take everything away which actually is evident from that OK what is that is what we are looking at now we take fiber parts of length of service to elements 3 elements and so on this week was the product so now was graded by deed the highlight is top floor number now let me a different look at Dick Marty the others and has critical 0 then if I said because I want to work so this hasn't has to a filtration system so then I can't talk about the normal functionality and exploits just taking some of the numbers and I can talk about the length of the field was discovered in the past similarly objects and has had only some most of the instruments 2 more hearings which means that more than 2 1 2 2 2 2 I don't know could borrow incentive so many given such amount excess I I can obviously better this product you so oversaw the generously a 0 we don't care and then sp committee composed like this then there was no they were I hope you see pictures of laboratory 2nd so but so then there's a degenerative fond of it you can define this by just taking the generosity of the objects and as a length functor just look on this as product so here's a picture the company but so wholly
optimist and so this year is a building in all origin category it has the top floor numbered 6 and output ahead here this is the B of the BI so now I give it did emerge index 1 0 0 2 so what does it tell me it just says I should shop that into 4 buildings and the 1st building should have top floor 1 the other should have just gone flaws in the other top floor too so that is a sudden requirement on this young namely is the knowledge that namely from there should be at least cultural 609 shop the building was stopped for 1 fiber product is gone for final .period gone for a final product was a thing of this this organization is important for the perturbations skewed because you have to wear work your way up as compatibilities on this so you have to know Hollywood chopper and wanted to avoid and so on exactly clear I would rather here think 1 of the best we have lots of time to think that this is mail the president 2 of the sold this year at all this is a building was hired 7 top floor numbers 6 so this is like the 1st picture you yourself except that so the protest was trying think about who is the head of the 101 to prove anything like this is delivered you know whether you will let me you will see that a little bit or a lot of people from each of law can be can many many things so but is it's not easy procedure just if you have enough of those guys the right number here and such a amount index the enormous read the length fall in London after the surprised 1 it's 6 which was of the number here so if the norm plus the lengths miners 1 it's precisely then you can shop in this way if I give anything if I get another 1 if I put 0 yuan 1 here than it would have gone for Europe than this year would be building of to so that different ways Oregon shoppers then the part of Beijing later which would see over this object who could be put back from this 1 and any other way was cut that shows a some compatibility sinks so no the notion of a covering from so so finally coming from the sun so you have 2 categories and although each object you have so it should be subject to objects and over each object you have like final number of tolerance and this condition should patrol which looks like the difficult at the beginning but the says the following if you fix the source of all morphism like a fixed so I am interested in this sport I take a point which lies about it and look at all and offices which start from such as this 1 this 1 this 1 and give the identity then this should be by objection to all the more prisons which you see here the original compositions and that ended with the social the government has the right to be confident that the source the sources and was with so this this Arkansas this album offered this is amorphous and settle this category this is an often settle this category that it has a source not into Jeanne effort goes into the objects that takes the form of prolog here and soul so if you fix if you fix this .period intake of premium channels covering then if you look at automorphisms starting here then you get a morphism To the objects starting here other goodbye ejection was loaded onto what is said it was reported that the government had to have the answers because so this is maybe I'm too the didactic you so so so normalcy with be objects and boldface would follow all face of often the 2 resulted in so forget about the definition actually we already saw such a fund to something
like that In the sell-off under the heart and soul take any accounting so so we have our category as we have the degenerization index obviously evaluation so now recovering structure for the sponsors of the following injury the it's clearance from intelligence critical 0 F j takes an element of the generous plus J. plus 1 and shops at precisely the right spot to produce the ability this top-floor what I end cultural project so the viability of a certain length second-set shopping your choppers Urich shop at your so and Archer ,comma informed just Texas definition from year to year so that the sources of his father definition so you see already that in this formalism keep very much tracked all the isomorphism Smittle portend for counting later on the generosity a number of however the generosity is the top floor number using European so it's 1 lesson what you would say the height of the In the end no so so then it's clear that if I have something and much of it here and there chopping there could also Chopard from their own children there which is precisely that so you have a company so chopping if you choose the right in this system the social barriers commutative so so then you in this rule would actually can extend this chopping and it's already the picture inductively for every monitored index so there is from as persist this product that there such a chopper so in that isn't
precisely this so if I take F 1 0 0 2 0 it is on S 6 and chapter said in this way so this all-covering fund it also establishes a can check if you just use forget anything I just under broad resort covering you created created from the smallest of coloring funders namely those guys you know to I think the whole thing has fielded no there's not that that's actually presently not just to the north shore 3 of 4 billion industry and Chopard here at the negative bitter 5 1 5 2 and then shop at you a good at 1st shot here and there so that's the size of this so actually they need to on the 1 hand and near sewn doesn't sing it should do so this year takes the right spot in between they have to cut that on 1 side and top pros I other jails and it forgets the precisely the head only the 1 at the right spot which is determined by giving I'm John that is what adults you don't change the object not just forget that for the you know and that is obviously rather commuted to forget the confidentiality of but it so so if you if you look at the data absorbed cases see that immediately so Is this if you have a building where each of the functions you this is the jail but if I forget that I'm not doesn't change and there's things they were more that's true so now now we come to some formalism which actually quite useful so if if I'm given such as Facebook from to and then let me define the boundary of this is a restriction to the boundary of the category so now given this structure he'll consisting of this
Maps here and then when I can go out of this I can't do it I
can define the cool thing here give us a little bit complicated and try to understand this sit down it is doesn't more than precisely that this is true for the seat forget the gym is just a compatibility that if I put it back c . time Seidel by F J I get on the corresponding us on Japan's 1 so so this saying that this is equal it just means compatibility things so if you have a seat you can obviously define a seat on the fiber product to just takes it over the final product is a problem of the seat of the Pentagon so to and then so so he is not that he is a top leadership pp for years should be complete then you put this back so you you get to the boundary of the category that this summer you and you want this condition it just means that F i j compose receipt of is just Utah restricted to respect type of Japan's 1 that is what it means so there is a shortfall due so is so so this quality means compatibility is a structure so so this is part of the conditions from which because 0 comes later so it was complicated but but really shortfall to fulminated while was have to say for J this is home to just say that now this former cause a lot already can threatens to satisfy this form and so I think he has to have some of the soul so moreover saying that you should be using the stopping who just below vis-a-vis takes values in 0 1 national just to the alternating song and that's a definition and what you claim but you require local would say compatible this is so this thing is defining the boundaries because the FIS come from there and as and as early as I rise at least 1 so this year defiance the new fund .period on the boundary and you say it's compatible if this is true and this is an untreated condition because they look at this but you take 5 part of Lang's tool then new restricted so that he does give the new restrict then you put it back to get something on the boundary and that was a supposed value there and you work your way up so there's a lot of compatibility so that's that's it but it doesn't seem more than compatibility was obviously FIG so now traders said another
many do about a Bayesian theory later awful for the other box structure may have a space out of some of the mourners of Jayapura Moffett objects and unmonitored plot on each level of port stays the same Jayaram off Excel and that I get a new moralize spares and obviously it should be the same to that modifications should be identified details yeah this is the yeah knows number 1 I haven't explained the pictures and stuff so when men so then I do a perturbation theory and for example this building off hired 1 would a more often then sold so as I would hazard moralists space of this kind of thing said if I had not by some Linda that should also be Memorial Estates because otherwise you never get the eligible structures that you have so if you only from multiplied by several geomorphic solution that should also be enormous busy might have different signs and so so if I integrate all that money was because of this it should be integrating all this moment spaces the staff to sign books and when was and so that something like this year at the same time with time on their laurels this time there was a sense that 1 of of the nation's if this is true it just means that for the tooled pro for foresees product of 2 things this includes futile but this plans to post bond for all idea that is it's it's a convoluted way to say this all at the same time but it's a good the mind like yes I am there's something to do with this supporters have a function to find function defined here and I want extended over the supports efforts to use users to you this asshole well justified in view of the cluster of Chilean miners jumped 0 you assume that they coincide here but that is a smooth extension of this this will doing the opposite so so if you do this now for higher things get is alternating sides so so that is precisely to see if you have to see the paedophile there's a lot of points that have multiple definitions coincide so with what stuff look so that's it online what so no truce in things so 1st of all they there so let me 1st say 1 solitude building would be 1 so if I have hired three-year would you say something like this where top and bottom half same patriotic orbit and each of them is from topic true the standard genome of excellence that's interested in the building and that the Jayaram off exceeding the building would be a major this removed the a lot of questions in the event the year award and if they don't building is 1 this 1 on each piece of so if the job His 0 1 and that in the product so if a product and 1 in both of them would be 1 them this is the building about the 4 that comes from would be 1 In his number 1 and that is not that would be disliked so "quotation mark so but In the perturbation theory if this is a light object then I added Jayaram automorphic silliness to the Sinjar Moffett chosen and a and it should be light as well so in the perturbation if I like this building so he also let's say on geomorphic cylinders and now I am a general Moffett building to then this was like this should be like but the same way that is otherwise you never get any eligible structure was a condo multiplicative structure the leader of the dead and in that they are different but it's something they have to come to the the same way out to sign a up to some so how can you
formalizes if I find of upon differ by Jhelum offered cylinder building then the W coincide so here you have to notice that there is not a unique way to put the quantum optics mandatory because you fix the currents class uranium on it and it can shifted to those on each level at least an hour Parliament so that is indicated by this but the thing about understands all necessary editor Moffett truce and 2 but that there not a unique way offended by the number that is just explain I just said that you had to order for the quorum more fix to the solution of building the true forum of inability this season he has said that I also have to a cylinder building Jamal Lewis said of the sale of its just line if this building has hired came it's a line of cases the job of unlawful entry yes and it but there is not a unique way to added to it to get families for each level that firm often so that you have been the end of of that's the young so is is shot way to run III editor traversal ability to work and then for that plan any representative of this because not unique the values should be the same right Heinz the times said it were it was not the case yeah I was so young so if this is the if there is building of hard-won 1 you have just the 1 barometer affirmative Boulevard tool for each of the singer 11 off so I'm just keeping it simple but so if I'm going to let him the but this you know what this is all about are so so he added so I would add 1 nobody even it's already not unique because you have Clemens class movements so you have to choose a representative Pakistanis and divide or by the action and again offer then have another level and was the same but it so my is that's the Dutch have this property there if I thought contains attributes Olindo building which is not necessarily horrible but if you don't on for his positive then the in a compliment all Jamil so you on that level it's a little bit more complicated to formally than later on a linear rise level the connection wants and domes and fountains and it's true so that the assets the right to receive the meaning remaining just 1 she was said right all all has many can do it again and put another chained to end so if it contains a reversal in the building which might not be but if it's light then the in the Commons all :colon was competitions which is yet you used to think of ways watch a lot of collateral the new artists from the In the early briefly according to handle it I know what the distance from me so I have the time or aspect but it's so that just so you know I'm using a series of Joel physicians and could not begin so I know what it's good but it's only you give me a few minutes here and and stop stop now that so if you have 2 nontrivial components each each building level has anointed the company that does not true and then you can show you can move them against each other creating again a family on each level often if you have several components moves the dangers other than what you wonders if you have to stay stable so if I find of a promise stable and you take some of this union of cities so on each level you're allowed to ship them against each other then the seat of our be the problem now so you have to think of this seat incentives would say fine Moffett things durum off again is disconnected on into then things individually already existed hospital so but it's so summary of what it's all summer but so the so we are interested in seat of dollar traded is the also at identification of fun don't identify Geronimo objects and 1 too the former two-seater the foreign properties this is true then this is true if all the ad just attributes and home offered said in a building if this is positive and this has to be true missing action had to be destroyed amorphous and the 1st ever since the meeting of a vision what this has to be true so as if the other Barkley comes from the that's all I think this is lower than the same old thing about the data 1 1 thing that you're considered to be 1 Arkansas the let me not buy into the so that it has had the effect of which I have to look at after that the and so so the IRA's alluded to in magazine orders multiple protect soul it said the IRA's but precisely that sold in the 1st the 1st time to the top you have disconnected objects so in the hierarchy concerning some complexity of energy on the bottom floor testings can degenerate firms their disconnected things this evidence connected which appeared to that comes from there I was the 5th so that is what I want to win the 1st 15 minutes to this has already requested that the I think if you write down everything last year we will launch will manifest satisfied by this the point for the but if you own more quickly than with yourself and you can find 1 things and we also will be unique to the point directly to the meeting so so if you if you take this property fuel used to Nunavut and satisfy weapons were removed last summer some people I think the real thing you will meet in Paris on the loss of that's a good result so my mom as we will just a minor former of cosmetically forget to the very end of the month at this point what we have done this week I have so far In this country were setting said what should do work so now the next thing is that the notion of a small structure on a category but you can only do with this fall and they have responded because of all the things you can do worse than you can start talking about snores object of scorn and news objects of this kind it could serve as a sort of small some the categories and what our so they actually occur in different flavors so many photos all the photos branch of reforms and so on and In this saying standard reconstructed by on soul so you can France processes properties into the language of what you should do on the perturbations on the linear and there's a nice from Torre description so this is obviously softened toward this is a little bit odd but there's nice formulation condominium for others and then you then the define a bundle of esthetic which amends section find anyone to perturbed that 1 and then if you will that is how it looks like at the end so you have so this is basically over each object he or the fiber is of your 1 falls among the underlying them so you have a question among from and then you define something like this Facebook saying here it's it's but it's a section it's on the fire was and his associates to each fiber to a certain number of points Russian number adding up to 1 and then this year's used the if there's a problem child if you put here c . 0 which just is a 0 sum bitch just give the red 1 to the 0 section into everything is 0 but this you is actually policy .period J. trade so if this year's survive 0 which just puts rate 1 and 0 vector and made 0 in the other than this gives you this country was giving that 1 majority so he was a small perturbation of his and in the 5 everything adds up to 1 what do literally Doris you take a potential unity for you use to judge a true for each object and then you move the objects of several wonders vertebrate in different directions and the lineup too many for his log let me through last year had requested only use around us and we will have to come back here at 11 minutes so thank you and


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