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1/4 Polyfolds and the construction of Symplectic Field Theory
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Title  1/4 Polyfolds and the construction of Symplectic Field Theory 
Title of Series  2015 Summer School on Moduli Problems in Symplectic Geometry 
Number of Parts  36 
Author 
Hofer, Helmut

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/16294 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  Topics: Short overview. The category of stable maps. I/O structures. Covering structures. Additional structural constraints. 
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00:02
Hey all the time and place the and so if you want to take notes I would only write down what you don't see the work you sort this this and business actually was the runaway started in December 1992 I felt that it was in Basel the or remember that there could be so regulatory boards once conversation .period and the National running through balls and I think at this point we already had an idea of what contract was due to commence some time grocery no he also so it can so as a certain number of issue was 1st offered something 1992 when nobody knew even out of the woods tonsils Armitage's ,comma written and so on and we knew that we should perhaps to some version of cylindrical but it was always an uncertain that's not particularly satisfying so so than in 96 and no member of certain solutions proposed for during this Council's the issuers and the thing at that point it became clear that in 1 way or the other you could do worse things but it wasn't so clear if the methods would actually 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 tricky so there's a lot more to learn if you actually go into details so that ultimately but Mendes was on a lost soul you could think of the home extra things have developed so this is a starting point supporters but other difficulties wouldn't be there it would just be a Fretilin problem what would have happened what it would just be a classical fertile problems than people would have their with like that was friend so was to take a small perturbation and at the end of the story wants to put that into this framework and then of course sudden sings would have done that you don't find in literature namely when you have to do other but you have to think about this then he should do but as if to you on this floor top series they then you have a lot of interlocking all the pending fresh from problems with capital Putin departed lower than the inflation among other new phenomena problems have to be fulfilled in the very practical reasons well and so on and so on so so so I think if this 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 which would show all venue would actually applied as safety to so that became 1 project in the northern part of the project actually providing tools to put a concrete problem into the spirit this framework and the report also model for a character it's so this was a solid scenes which are nearing the end namely 1 of the annoying features of on when you read any papers that construction is usually are rather global so everything is interlocking only analytical topological dramatic hospital on this and then you know that you have to write it down you have to write a lot so it's tempting to say 1 can easily see losing standard Nestle that that can be done and you have flocked to some people which refers to another paper in the form of the sponsor clear reduction happened so so so therefore what what what what in principle of anything you look at elliptic ceiling so you can make a local estimates and just clear covering unit of estimates of this sourcing work so sounds and the ones you want to have this kind of things here wrong so that also remarks on the market so then ultimately the was almost forgotten is that restored from a problem which now from this perspective is actually I think rather transparent so and other talk about precisely how all to do as if to assuming that you would know this is missus plentiful language of electoral explained the things which you have have to know and so then of course because of the lastminute so many causes of central government deal but about sort memos ingredients Crockett said so here is a stable next so I want to expenses but understand what it all means so this year is agreements office this up opposite things up upon the buildings now we use European notation for the floor this is or 0 because certain things later should have degree 1 floor 1 floor to all 3 so now then there are marked forms like this many this arrows here so so this shows you tall half lines intelligence space so if if you have a punctured and oftentimes space your fixed disk then but you can put corners on the disposition of 0 to standardization of 0 to this point but many of his life you can have 1 In the erection of those lines so completely determines what you have to do now he'll however I'm
08:13
looking at is to arrows is up to ,comma notation no you but you look from right and left selection wanted like this but so you got up to competition and what you want is if you take Porter Corners which are coordinated this way and you run along city on his 1 0 or the order action positive or negative that hit the same point on the periodical so I do not so let me say that to the to the as of TPA parked there are some differences in this case for the transfers nullity I do not None of the pawns I do not numbers of passengers that gives me a lot of other prisons which I will have to be compatible a perturbation have become part is as large as that of isomorphism if I number of them then if I take it we numbering should perturb in the same way as a little more complicated keep track of source if you just throw everything away and utensils arguing that level that's 1004 intention Christians and you might go to recover and ejection precisely means putting numbers so sudden that you know you have so you attendance Perez who take off a beach towns inspectors look at half lines and once to fix a half an hour and then if you fix a disk and intake Pollock on that's a decade this level composers into the sums and then if you run the and and it wouldn't Inditex adjustments such a way that really positive line In the tunnel space is now up to that line that fixes besides were caught so now but we don't think but we so onto adjacent notes but punches and they are allowed to retain commonly but that the effect of this is that if you fix so coordinated system of records 1 positive 1 negative thing if you run along 0 across all plusminus 0 and 0 minutes 1 then a day hits the same point on the so now if the Perrier recalled a simpler covered there's only 1 shots but it's not the call would you have with this 1 but it can also take this 190 degrees with 32 that's the rotation of so so they are actually came in choices of the visit is losing its complicated for the same thing he said that the mom and allow it what we you want the 2 common the idea that is what is because you have a commutation divided our private .period so but I have to attend modification by budget the other by because of positive narrative so most of the time was last year of that this doesn't run on bank I'm in the middle of torch so so and so then he had his newest of is recalling ordered pair are so need appear 1 that go to a different point of Obama's sales of the other which you should identity single for being identified so absolutely continues and then he'll inscribed reckon that as well close to a conference on that and business on August unitegroups aluminum just maps and then this crab record around it means it's in our colossal recalls the image lies in all across the state of automation benefits of his statement on structure and 11 are action soul if you take the map defined on this things here and you just shifted by hand or by constant it's also so it's incumbent so there's a the we also said have a were most of them no not yet at the moment is adjusted so it's really Minnesota all always features then the court so then of course that at some point after the summer none of this would have to be a little bit specific about what the map should do value so the quality of the method chosen so that it can do someone on so I of picketers H 3 lock nonlocal at 3 on this end user punches but they should if you take port cornered should be of Class H 3 days coverage of scenes of his exponential rate and the exponential rates pick have to do is have to depend of course on the periodical and in order to choose such possible rates you have to fix an almost complex structure that gives you linear of of other tried to cooperate on the development public operators Savage and on the generous Suncheon means that if you look at the party conscript concerned was a context structure of his please probably abundance there 101 cases doesn't have 0 in the spectrum so each of those guys as a spectral get and you have to fix it and then of course if you sincerely generally have a limited Roberts the gap might get smaller and smaller Hugoton Mulholland appointed Roberts sold for each of them you have to fix something the spectrogram wait so it was clear that up to all translations if there is 1 of the are occurence Crosby this is Sarah Hamilton but many farmers there structure so now there's a stability conditions namely leased 1 so that each of these things suicide I think of this not be identified the remains of each of the sayings of you as a complement of monuments of so that if you look if you're going from 1 floor 1 each for all of that has to be at least 1 component which is not a trivial Salinas was a trivial Selinda attributes and there's something like this year which has a positive and negative function which NBA AllStars and taught it to the same periodic orbits and homotopic to standards Jayaram offered said How much of it too the president has no indeed not but is entitled to the same thing so it's hollow topic to the other Monday into Donegan it's 0 so I write to Italy because travel a ,comma you'll ACR components and the said this is the stuff that indeed then fall connected components of sea but which is not a trivial Selinda like this 1 here we have 1 of the forest was the integral only us positive or even 0 then you must have enough special points on see job ponchos marked .period snoring in Poland from the normal appearance is a clear what the object so this is no object in a category so so this year explained so but so this explains the following if you want to be precise you you have to say that this Pancho corresponds to this punk trying to have to say that you have this lined arrows to align hours of course you can give that by a complex I'm telling the Honourable complex in amended to change the structure on 1 side 2 miles apart so so there a good thing if I look at a ponchos and I takes attendance barriers then I can take so we'll put rectification of this it's 1 government naturally but as natural as 1 inch on so you have something like a it principally as 1 bond over the pensions and you want to have a complex and you want to have a "quotation mark if you take the take
16:23
you the complex structure Jamesview minus J U 1 driver but as 1 bonded isomorphism which gives a bijection between the underlying ponchos and it says which line corresponds to what I was less than a hundred things and jails the sponges but history I don't know yeah itself so but I don't know what you mean by data could possibly to a different countries that could be jailed the target that could be J on them OK but it's so when our kids so I can you would just give me a little bit of a chance to to explain this that's what had been a precise definition but it would be good before you see letters in talking about it that's OK thank so now I yes you can but the union is a chance to finish this and after the so so there particularly so if I look at the Pine would be the positive function of the floor "quotation mark minds 1 to the floor they punt on the floor fluoride than projected was the tangent space real there's a structure here was a minor structural and this will be the principal bundle as morphism covering this projections for the bijection would say which puncture cosponsors what and this would align the Arabs said explained let me say this assignment who was accused of countercultural there was wasn't so now so now this saying which you just so can be encoded like this this of course is the story of a sequence of the art of fires is 1 floor narrative punches on floor number SIR being surface and JI to J. on floor and I marked poison for DI love of Nordic appears on floor so that would be a horizontal things or flight the equivalence classes positive punch then this 1 are precisely the ones which I explained here which could territory will hold 2 lines it is time to cool Poland and have I suggest told you shouldn't because this is online no country and no country this is or not just open your Singh and right on 216 so now this is it following good although Udinese who manages what a man who has been explained it this way of thinking the a which function from the law floor goes to rich and above is alignment of the stars of the hit back the identification this tangent space at the functions of the service for this structure naughtier is a minus sign for the narrative structure under the principle of London principal 1 as emotions the way in which the EU units it is this this call that all right with me next time you see you but the fact is that it be the 1 who implies the use of the previous and that was only about half of the yes that including the rotation but which which lies on up to 1 . good so what is the morphism so much physically so this is an object my category some offers and I ride like this there is a source of target is a sequence of but by polymorphic maps so they get from here to here which means that many ponchos compunctions just minute on this spot viewers JI has to be biomorphic In lies being up to a month prior to the methanol appears and so on and it should hold that if you wish there is a constant on each level if you shift you you are and then you get the composition it was good to see you again North Korea has changed the yes that is that that is on the sofa each floor you're allowed to common shift of everything the despite adding a constant your company because we remember the objects were curious glances and so on so this automorphisms Americans have reserves this year said it yeah you know or care yes it should so so if you if was good question so you have to fire chief II it actually that gives you so it is that it shouldn't happen current matching tour terminology but actually quite important so for example this to answer this question of illustrated more so if you have but some supporters who have true daughter ,comma and also very considerable 2 1 1 treated other counseling then so it particularly in the middle of a said of a covered and there you have as matching influence so now this can be isomorphic it can be it can be isomorphic to Worthing which has this shifted by 180 degrees but 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 what is not so clear about which is the consequence of the stability condition of exercises that to objects only finally Romanian orphans now the soul of course according group on some employers you just more category which would have this property according reported because it's not a small small but they in this category in it's degrading because I can talk about the top floor number so if I just have a building of all the building of Hyde won the grading would be 0 that means ground only sees a ground floor of building of told the D. would be 1
24:33
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 physical distress but this is isomorphic to a different object like this 1 that stays with tainted by 180 degrees but that is a different object different but it made its something so they it's something like Facebook like or not like but was Russian confusion saw so so you would have had you know you can kill plants would 0 1 with just the identities of morphism summit in front of them and you are interested in and and if I am given such a thing I .period category which all the things which I like to some degree political and they are have carrying away how much Eriksson like you some examples given J. anomalous signed an almost 100 such art was made and a compatibility as loses their structure we can talk about jail anamorphic object the nets which other you was because objectivity was that this was a crucial immigration so then defined this by associated to Rome after the raid 1 I like and to this thing I don't like them so that our model aspects for the time we or sold to another 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
32:47
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 socalled 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
41:37
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
46:24
like that In the selloff 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 topfloor what I end cultural project so the viability of a certain length secondset 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
48:21
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 allcovering 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
50:43
Maps here and then when I can go out of this I can't do it I
50:49
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 visavis 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
53:58
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 threeyear 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
59:38
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 hardwon 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 twoseater 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
00:00
Point (geometry)
Classical physics
Building
Multiplication sign
Sheaf (mathematics)
1 (number)
Kontraktion <Mathematik>
Canonical ensemble
Mereology
Perspective (visual)
Number
Explosion
Frequency
Mathematics
Centralizer and normalizer
Positional notation
Manysorted logic
Causality
Reduction of order
Arrow of time
Subtraction
Units of measurement
Series (mathematics)
Standard deviation
Spacetime
Projective plane
Analytic set
Algebraic structure
Line (geometry)
Perturbation theory
Mathematical model
Bilinear form
Differential geometry
Estimator
Field extension
Cylinder (geometry)
Network topology
Resultant
08:11
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
Wellformed 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)
24:33
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)
41:34
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
48:21
Building
Sine
Algebraic structure
Algebraic structure
Functional (mathematics)
Category of being
Formal power series
Mathematics
Funktor
Computer animation
Object (grammar)
Boundary value problem
Condition number
Object (grammar)
Nearring
50:46
Point (geometry)
Building
Greatest element
Ring (mathematics)
Multiplication sign
Mereology
Equivalence relation
Event horizon
Theory
Number
Frequency
Finite element method
Cylinder (geometry)
Sign (mathematics)
Causality
Object (grammar)
Isomorphism
Boundary value problem
Energy level
Cuboid
Fiber (mathematics)
Condition number
Family
Standard deviation
Multiplication
Product (category theory)
Spacetime
Process (computing)
Building
Moment (mathematics)
Algebraic structure
Staff (military)
Perturbation theory
Functional (mathematics)
Automorphism
Orbit
Maxima and minima
Category of being
Field extension
Cylinder (geometry)
Object (grammar)
59:36
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