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# Moduli Problems in Symplectic Geometry - Polyfolds discussion with Nathaniel Bottman

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00:03

boy the plan had been for helmeted discussion but that's the shifted to Wednesday but because organizers felt like we should make sure that everyone is on the same page from last week so that people are just not getting anything out of this week I'm so the main point of this is to take questions from you guys but I thought that I'd start off by calling for you the definition of powerful mostly as a way of putting a dictionary on the border between some colorful concepts and classical OK so here's the definition of Polly full of holes all yeah I'm the Polly for specific terminology so Galant section strong model why over I too this is over X 8 came and only fold is still fragile I it the following editions holds of the 1st condition is efforts regularizing and the 2nd condition is that it has a film version which up to a field was nations of the following 4 because of this we want to do remember yes thank you so say thinking so at age group .period or nearby .period you guys trying to get a slightly imprecise version of the story I think that's light on the way up to scale plus information is the following OK so Our bundle it's going to look will look like following so W some scale space mumbles locally genteel Southern bases are a little and W and the fibers are big and plus W and this section is supposed to be what was the principal part something called and I'm on the right worldly what property has satisfy services like that the offer so that property is that if we look at little Joe we subtract off g at that point were centering on and then we post compose that with the projection down to W OK so that's the W and the fiber and then we apply it to a point the comedy you so little of the lives are little annually and W that is of the form you minus b of Yukon the where he is you can think of it as a family of contraction mappings franchise the are which were reviewing the survey's living high-dimensional dimensional space family contraction mapping sprinters and specifically what I mean by that is is that I make any choice of em the capsule the following inequality holds you a new Prime closed presentation of I'm close to 0 can depend on the choice of eminent so it's generous definition but the point of the writing on board is not for you too please understand that right the 2nd the point is to remind you of these words which French last week and that's when it all the time and Polly fold regularizing the old they apply that's everything and now all recall for you the dictionary between those terms and concepts used yes do it have on his head in the loop to buy this is this is stated in the boundary in this context education said let's yes there is a version of this advantage so we don't have too many the concept of the same time let's to the bench the you know it's is

07:02

a it's an R & R yes yes tradition .period they said the but this will be true smaller and smaller neighborhoods observe also the instructional children all of this was of those so this is not yeah and dumb if there's time at the end I will I can motivate this because of similar properties satisfied by classical French maps and in that context you can think of to be something crew differential vanishes and therefore that's why in that case the quality satisfied go on small balls were the smallest and found that that was going to play this year because of could you the hosts OK for a variety show any questions about this statement or complaints OK dictionary of all right here's the scale setting in years the classical version for the 1st word St plus when you see as plot or more generally when you see as you should think of the Infiniti and seek but when you see the words Strong bond you think of a bundle where the notion of combat perturbation makes sense OK I let me not pertaining to this dictionary when you see Empoli fully should think of a bond not not but when you see SE plus should think of contacted and the only thing I laptop feels that it's not exactly a classical notions and lyrical fear that what is filled me in world priori section F is defined on an open subset of skill-biased spaces defined on a retract sitting inside such a thing which is no not such a nice space as far as produces thinking space dimension Avery locally and so forth and therefore nerd and people from theory you have to be up to map actually doesn't mean open subsets of Obama space and that's what this field section it really yes questions Oh yes should should put that year regularizing the integrated so the 1st section comes from Delmar operator the regularizing property comes from elliptic regularity satisfied by that don't bar operator yeah I I don't know ,comma tennis but the 1st Commission that I still retraction but say or which goes from next exit still infinity and retractions then on the 1st condition emptiness is that the January see Annex of Oravetz is equal to the degenerative next of access for all Texas again so I don't want to really get into this because I think it's a little bit more technical than the rest often the bird it was stated that forced mission correctly it was the 2nd edition "quotation mark the intelligence systems installment and there is a conflict of interest in use trajectory you they should just go along with so at the last minute changes there into the evangelist setting and therefore I could have erased the gains In the vulnerability of the status quo we stones Sean the following instructions from the launch of the shuttle somewhat the this 4 jerusalem it's not expenditure because we were traveling at usurious loans to reject of the world obviously the condition that thinking of that

14:03

but this is your season you know what want the Museum of the Moving in the opening the maker of various forms of the rights of the point is that if you look at this point right here than the the generous index of the retract his 1 but the designers cynics in the end to so far the nevertheless this line has been used this is 1 of those companies OK so I suggest we leave it at that great welcome so we have this dictionary just bring everyone back to speed up and the last thing I want to say before and then move on to whatever questions you have is that the most basic reason I mean the real reason for using this walking definition of analysts is that in the classical setting the reason that all although their injuries to 4 fighter dimensions are stated in the box setting for French maps is that the satisfied this this contraction normal form and therefore things like the inverse function there you want to use that Hoddle said but there's some problems with the levels so that if you just assume that your as operators girlfriend home systems you want the true like the implicit function would not beaches and therefore builders directly into the ocean like this at all come back to this at the end of the tunnel but I wanna stop talking about this for this property now we have the feeling of relief we think this is the end of the 11th does the so for every element epsilon there's an open sediments inequality holds on the end of the the slaves and they will love it OK and so me open up the floor for questions which I answer or deflect "quotation mark so this is the people who serve you are all the schools world realization of the constraints of their security generally is there was a lot of sections the utilization usually does not and continues authoritative on this 1 this is the loans on the books change the merest most so socialization can be very moving images over that accusing his of its general this training was saying as little more hours this is the job of the use of a single through because it all in the past the grand jury arises on the durable solution and so we don't know what the solutions that his projection from the status of the structure the chosen to yeah you don't have been in that it is in the yes that's right what I should do if you want to make some money challenge shuffleboard In right here is that implicit function .period OK so let's say I would say that why over access to strong bottle and was just point out beforehand with them that this is about listening but for some reason you could retain their work so that a strong model to the strong 1 of over X Empoli falls with no OK and after the steel from section With the property all of those years Asians are on so such that for every exit said so wanted all of the victims we still have a long way to go some of things OK I can attempt to you can correct me so well it's a solid say that such such that and at this particular point X not the lean years ations suggested some Act dating back to just a 2nd so this is supposed to go from the tangent

21:03

space X to the fiber over the fire over X knocked out so there is that there is no existing open said history might make a mistake of topology said exists you an open so that the incidence level of X containing acts not with the property that if we look at the start of an intersected with his open said we find nationals of employee fold and it's there end the coming the 1st automatically the structure of high-dimensional infinity manifold this is strongest the presence of women at the top of the mortgage the result that it's abide by the regularizing properties .period because Napster 0 1 I'm sorry you're saying that directions to the the of while found in so researchers say again it's all part of the spot to 4th in the utilization of solutions it's which is and proved that this is so that he is still so there is a small hole that were solutions that as part of whose was also sentenced to years additional solutions to the absolute limit on that point so right and for all of the facts in the year intersect solutions said prime at X is going to want to use 1 as those the book the sole authority to select his father grew the only was the source of the but some people don't want to so tell me if this is what you just said the school direction by police you also want you think it was possible that the the result was statement it was not all meanwhile the innocent here's what nonsense to think that the use of your government and the resolution of the problem here is that if I wanted me to on memorials is it's still in the strongest they also uses the metaphor of interest from this emergency focus great so as that and your question here is who is going to make sure that all of this .period this is the 1st of May the force of the kind of thing that you don't know where this month In the image of the structure pull allies the size of solution Susan at the idea of proof of the imprisonment of their interiors is using exercises construct such an hour the composed of Europe and on the right so the image of the tension are equal to the Colonel OK but I can think of flying have this year I want to read about light years the values that out interviews the discussion with all this is all you can said the retrial news to the North's view of what's new about this region resources from the indian this was responses to and then announced years was very euros most of the them problems with the size of the their so in this case the role of the creation of the appropriations bill would also of through the I think what I remember is the solution for Kosovo job is on the world to the right this thing is going to have a way to check the the quality of care the but what was going on the fact that you will surely won't work because it through the history of the world will see considerable also pointed to the role of the so I didn't understand what he said about how the changes in the evaluation that in case was built to move them the series of sold the whole thing going on because of the FAA wants to use the common points so that the results were reduced by the time of the hearing all of the time so you the that's going to change the scale monasteries working right the conditions here nodal point has to be in common between Moldova and was like Qatar as far as soldiers arrived in the U.S. state of all the bridge that is 1 of the Group of Seven Years in this case is the the source of the strong constituted the fish farmers in all of you "quotation mark

29:52

there goes a long way to In addition course so we call this phone so you can look at it more as a result because you have constrained in the use of the euro as do only receives horizontal or that of other competition some of needs of on broke down so fugitives Phil Reynolds sections and another was the people the latter case this year can you control 2 of the Federal Reserve will oppose plans I think we just like that is not the end of the study however raised the and then decided that it was the humane to rejected so that you know that the same and will support situations for example China now with the the police were not just the justified 1 from 1 it was the and so this flood product good go all the way what we are with you section he was the have 1st of all the name essentially system of the latest security from the region the quality of if you went through a whole set up the 2 operators because he not only is the only 1 thing that's the thing alternatively movements in the attic what is the best choice you could also say that they have been mobilized the quality of the United States will not only have we heard to and from the parable of the good things about him I think that would be it is exactly that police I strongly suspect that someone besides hung with Christene caption as some basic confusion from last week's site when I encourage questions like that anyone In the yesterday we wish and rights correct and this is going to sound trivial but useful to note that if you let's see we start out with troops shock we start out with scale from section after which we don't assumed to be subjective anywhere but then it's clear that the field section so the filled section means put together that would be isomorphism that used to have between them the complement of 3 tracks the text and that wide so that we have this still section of public capital left so it's a it's a fact that this filled section which is now going between on a scale 1 the open substance there of has classically French leaders station yeah isn't known I'm sorry I yes think you but when you spilled from home the new stations that every X in and X infinity and home arrived but the Fred home index of pop but has a firm next OK and let's say that the the index of this lionization at some point next not is equal to you I then I'm then you can't 1st applied at the same that you can always perturb using scale plus sections to get this Rosella dissatisfied .period such I think I might be mingling their it's a a hole in the wall of 1 of the soldiers on through the reason is that the euro little jewels the debate about the use of the fact that I should have been clear about the local bonus of them were more so that the thing that I wanted to get across and clearly I'm not saying this correctly is that there is the notion that you know there's a notion of from index and you know even if you don't sensitivity and you can't perturbed to get surjectivity and that the dimensions of the financial standing in for cut out is equal to the phenomena to the original so rather than try to make a precise let just erased this is the thing there have been a bit of a former review Over the past fall we years was on the so solution in the short the division of the government justices of the existence of None of With the United States in applications was 1 example of the people of force also almost American relations using the question the 1st for nearly all of usually it's the idea being that like like I mentioned on Friday is the setting for cylinders or something the asymptotic operator if you look at you your operator take the limit is you got to customize if indeed that asymptotic things can be nice office and that's used to find the the prime hold but then

38:41

having said professional journalists and very precisely this sort of thing going In there because the identification of the dead and the other is the only 1 of the lowest news saying that the sort of canonical traces corner but those coordinates depend on what you're good at it's but anything less than a minute my understanding is that the users are most natural way to prevent applications to use this alternate definition that the catch came up with so she came up with this definition which prove looks more complicated but in fact it is easier to use but I'm not the expert on this stuff Due to carry comments are correct but the question so I played with the 1st of all erections on and some uniformity in the rules of In good direction this that is so it's not sold a section of this real estate good to put friend brought and the glories incidents I want to go to work action is saying when she says that differentiation is I think the key 1 captain said is that when we're looking at is reprivatization action it was not differential but is not defensible exactly because of what was going on the :colon printers a different shade in the function direction as much as you want but the bad stuff was happening and would news flora the uneasy sticks the and brand that it's the new look she wants to buy the group has called for the release was impression was that philosophically that was built directly into catches equivalent definition of an honest this thing I just said about reprint position action picture and there's a really great right out of the proof of that the fulfillment of the Dover operator in the Hamilton in the case in protections on the archive the title but 2012 I have a lot of his 1st placings only the conditions in equivalent mission of the Constitution is the last thing we want something on Monday so I I thought I explained that the right to strike down and attraction I think the answer is no well what's an example to on the other side of the current financial year was in moment for all of us have to all of a lot of good question so so 1st rejection they steal attraction is scale infinity now or which goes between open subsets of scale Obama spaces we are on so skilled rejection that's it as really simple definition the mission of the spies explaining why the right down but it's a special case of scale retraction so in scale splicing and his map following form so let's call going from and let me right down the case without country plus 2 itself following form so it's going to send . ,comma the comic 2 ,comma the pies With following properties the 1st thing is that these pies and families of of the the retractions for all I severe which goes from the itself see when he was called steel projection gasoline it is safe to linear scale 0 maps from the EU itself which squares to himself In the 2nd properties that P itself as as a correctly the look so right so that's right so what but it's it's mood in the sense that we put them all together capital appears as a smooth but your question was if you look at these operators yeah it it's not even going to be continues as a map from or due to L of the common what were the only 1 knew that was that the attractions of the most coming constructively on can the conservatives system the men were called all right yeah that's your our rights so and and 1 is reminded that this is the example that we went through on Wednesday what this was the 1st and both of them so it's a family of Minya projections were this

47:29

aura of priori like you have no idea what kind of and yet there's not this party this off of me the franchises some kind of family that ones this is the place that contains the this is a time to be on the approval of the person who is similar to the it is about to find out about this rest of the the British out the bad news said Peter you're used to years of the properties which fall from what the role you I was it is not the way Solera copy that this example that I we talked about last Wednesday but where the retraction was on the market said are if example of this place so in this case and running the the horizontal direction and for any positive the projection Tyson these projecting onto a bomb function projecting onto the one-dimensional space standby about functions centered at the 2 the 1 for the lessons for equals 0 civilians and and as was alluded to every single retraction that has come up when constructing models the home of curves it has been skills so does the big 1 years when you're projecting onto the kernel of aren't that's that's going to give you the so whereas there you happy now that was the value of the injury I guess the Joel actually as an example ,comma called instructing handful of maps the other that that there might be some dinner the theory that it's not you always will be I think of all the more so because if you don't want to lose all foreigners of mortgages because of the ongoing on the phone and call the police for more than half the full development of relations between the 2 of them in the form of 2 now that the use of the forest the material more than 100 thousand in the hands of the people of the 2 of you know so you we should not because the people of much of the all right think right now so that if 1 question and I wanna something with them yes I think I have something more important to say if you wanna read about that it's like habitation along its super-easy it's nice to get a paper titled a general for home theory number to buy over the substance In the structure In the last question OK so why 1 doing that in the last 10 minutes I hope I can fit in is on is is prove a like the easiest possible version of a regularization and the reason that this is relevant is that the Pollack told version of this is exactly the same proved basically with stale stuck in front of some words but right such as it is lifted from captions costs a couple of years ago and right so here's idea so was signify dimensional vector bundle living over B In essence this section of it so here the defining dimensional manifolds finite break after all S is a scene from section and that is is compact which turned out to be crucial to the proof so what they're saying is that you have you have perturbations so conclusion is there exists a set called P sitting inside of the compactly supported but right Stephen these sections put the ball properties so OK I'm the first one is saying that there are there are elements of the and in fact you can find arbitrarily small elements that exist as sequence pieces such that PTI goes to 0 in the next 1 is stranger salad that's that's pretty important so for every piece in this current P uh if we procured key than that intersects the 0 said transversely which is to say that for every B in the 0 7 billion years edition is on to sell DVDs of AZT was key this is what has happened in space of the at to the fiber thank you yeah OK and the last 1 is that compactness is preserved gasoline her Fisher and it's pleasant and like I said if you look at the ceremony also have a preventable if holds essentially so let's fix we on in the suit in the solutions so while hours solution isn't necessary transfers that point we knew that we know the Corps colonel of the lionization is finite dimensional since he fired so then let's choose a racist the 1 3 yen for the coca and all of the organization at being then let's extend these guys the compactly supported sections OK so that these finitely many sections soup up our original vector so let's look

56:18

at but the following things sitting over the Times or so the projection of the obvious projection and now get this new section called best-seller as defined by setting as told applied to the X 1 3 excuse is defined SMB the plant x 21 been through exon team at the in August of course during the trivial things the Aga action now the point is that

57:12

now we've killed off that Coke Colonel Abdi so we know that as

57:21

told His chambers to 0 at that point the and it follows from that that there exists Delta great 0 the answer you which sits inside of being which is was the neighborhood of getting on with the property that it's told is actually transfers to 0 on all of you times the ball the radius of the center of

58:07

an appointment were now explained the compactness of the original solutions said Tuesday that we can moralist ,comma there sent by finds many of its any questions about this so far OK so let's cut which contacts by finding many opens since we can original process and not in the solutions OK so then without allows us us to do is weak although the using of Britain what the losses today if you break down the supplies is constructs a fattening up the time articulating over NYTimes . In this section as told With the property militias correctly right told his transfers to 0 on the for me on a set of 0 times of already has so it's now crucial there were only finally many of these sets oligarchs informed don't focus so were almost done so let's set signal to be right at the time the doctor 0 said the solutions told on I'm sorry it should have been efficient neighbor to the zeros some useful is to continue to say that now makes sense OK so soon because of the 0 set of Estel is cut out transversely on this I statement fundamental that's what was expected OK

1:01:16

and now essentially done because we can consider what happens when we include Sigma into the base and we reject down to park so let's call this map 2 and so that we can play starts there and acute note that when the totally fine dimensional said no problems with Serbs there To soften tells us that there exists some appoints and wine art is small as we like but I would say that that's what's can allow us to prove the 1st part of the so this is a regular value of and and so it follows that if we look at plus 1 1 2 1 albeit through blighted TK this guy which is a section of original but will you be His transvestism which is only 1 in the 1st place some that proves on and done let me tell you what you need to do to put this into the colorful setting so the 1st thing is that you need this contraction and part of the definition of scale from homes in order to be able to say that but you know solution sets of things transverse 0 horrified officials eventful and the other thing we need aura you need your stereo box faces to actually scale Hilbert spaces in order for some functions be it's nor will this August anyway need but functions and besides that other carriers any questions about the press and then you this this is the kind of guess so you can conclude that you can get salad arbitrary only with skill plus section this is the stuff that I this time in a year I I guess is a different kind of rigorous is the regularizing the sense that you end up with a smooth manifold we have I will stop now the agreement is not 1 of them In this you 2 consenting sensible I have worked up I think that you're just going need to you can start off with these things the transversely cut out point basically need to extend whatever perturbations you need to get chances salary into this whole co-workers and some of the people of the region we will respect the will of the people the UK the move because it doesn't have a lot of is you may use of extended over the same period of last year but insisted his the 1 thing I wonder about is the initial step when you take a perturbations anyone extender something which is not surly transverse but you certainly at least friends home so here so you don't want on the the mission 1 of the New York you because of we way the mayor of just of poor there is also means the reason for the longest confusing certainly you if you expected others homotopy through from home operators be OK but there but is wise immediately into that 2 France if you want I see I see OK right so he that we use this contraction for professionals in the world we and there all

00:00

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06:55

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21:01

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38:39

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56:16

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58:06

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1:06:42

Computeranimation

### Metadaten

#### Formale Metadaten

Titel | Moduli Problems in Symplectic Geometry - Polyfolds discussion with Nathaniel Bottman |

Serientitel | 2015 Summer School on Moduli Problems in Symplectic Geometry |

Anzahl der Teile | 36 |

Autor | Bottman, Nathaniel |

Lizenz |
CC-Namensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. |

DOI | 10.5446/16316 |

Herausgeber | Institut des Hautes Études Scientifiques (IHÉS) |

Erscheinungsjahr | 2015 |

Sprache | Englisch |

#### Inhaltliche Metadaten

Fachgebiet | Mathematik |