Moduli Problems in Symplectic Geometry  Discussion with Helmut Hofer
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Title 
Moduli Problems in Symplectic Geometry  Discussion with Helmut Hofer

Title of Series  
Number of Parts 
36

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
Identifiers 

Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

Content Metadata
Subject Area 
00:00
Complex (psychology)
Greatest element
Wage labour
INTEGRAL
Multiplication sign
Direction (geometry)
Orientation (vector space)
Materialization (paranormal)
Manifold
Kontraktion <Mathematik>
Number
Morphismus
Operator (mathematics)
Energy level
Subtraction
Observational study
Rational number
Theory of relativity
Spacetime
Projective plane
Physical law
Algebraic structure
Mortality rate
Bilinear form
Orbit
Hausdorff dimension
Phase transition
Object (grammar)
03:56
Complex (psychology)
Orientation (vector space)
Multiplication sign
Decision theory
1 (number)
Mereology
Sign (mathematics)
Mathematics
Manysorted logic
Spherical cap
Homotopie
Determinant
Position operator
Descriptive statistics
Social class
Physical system
Covering space
Spacetime
Process (computing)
Price index
Perturbation theory
Functional (mathematics)
Bilinear form
Radical (chemistry)
Cylinder (geometry)
Order (biology)
Normal (geometry)
Identical particles
Resultant
Directed graph
Point (geometry)
Trail
Chain complex
Observational study
Parity (mathematics)
Tournament (medieval)
Canonical ensemble
Theory
Number
Frequency
Causality
Term (mathematics)
Operator (mathematics)
Reduction of order
Boundary value problem
Subtraction
Condensation
Time zone
Eigenvalues and eigenvectors
Surface
Physical law
Independence (probability theory)
Algebraic structure
Line (geometry)
Set (mathematics)
Quadratic form
Local Group
Field extension
Equivalence relation
Voting
Universe (mathematics)
Fiber bundle
21:22
Explosion
Category of being
Beat (acoustics)
Euler angles
Mathematical analysis
Right angle
Perturbation theory
Heat transfer
Mathematical model
Position operator
Number
22:11
Point (geometry)
Building
Identifiability
Observational study
State of matter
Multiplication sign
Direction (geometry)
Decision theory
Sheaf (mathematics)
Water vapor
Mereology
Open set
Number
Explosion
Frequency
Morphismus
Goodness of fit
Latent heat
Manysorted logic
Natural number
Operator (mathematics)
Energy level
Boundary value problem
Length
Subtraction
Fiber (mathematics)
Condition number
Area
Series (mathematics)
Spacetime
Product (category theory)
Forcing (mathematics)
Gender
Mathematical analysis
Line (geometry)
Bilinear form
Estimator
Orbit
Category of being
Arithmetic mean
Equivalence relation
Vector field
Right angle
Game theory
Uniform space
Family
Resultant
31:52
Complex (psychology)
Group action
Building
State of matter
Direction (geometry)
Multiplication sign
Decision theory
Sheaf (mathematics)
Insertion loss
Function (mathematics)
Mereology
Weight
Perspective (visual)
Grothendieck topology
Explosion
Mathematics
Manysorted logic
Cuboid
Area
Spacetime
Theory of relativity
Product (category theory)
Topologischer Raum
Generating set of a group
Basis (linear algebra)
Perturbation theory
Variable (mathematics)
Bilinear form
Time domain
Flow separation
Category of being
Cylinder (geometry)
Symmetry (physics)
Lattice (order)
Phase transition
Order (biology)
Configuration space
Right angle
Evaporation
Resultant
Geometry
Point (geometry)
Differential form
Trail
Chain complex
Observational study
Civil engineering
Diagonal
Connectivity (graph theory)
Manifold
Similarity (geometry)
Regular graph
Number
Frequency
Goodness of fit
Natural number
Operator (mathematics)
Energy level
Boundary value problem
Function space
Analytic continuation
Subtraction
Condition number
Series (mathematics)
Standard deviation
Forcing (mathematics)
Neighbourhood (graph theory)
Mathematical analysis
Content (media)
Algebraic structure
Line (geometry)
Mortality rate
Mathematical model
Cartesian coordinate system
Quadratic form
Summation
Voting
Logic
Network topology
Charge carrier
Vertex (graph theory)
Mathematical object
Object (grammar)
Local ring
48:20
Area
Spacetime
Differential (mechanical device)
Basis (linear algebra)
Algebraic structure
Mathematical model
Price index
Bilinear form
Theory
Local Group
Subset
Differential geometry
Force
Tangent space
Field extension
Natural number
Boundary value problem
Bijection
Untermannigfaltigkeit
Position operator
Chain rule
Sinc function
Social class
52:28
Classical physics
Point (geometry)
Series (mathematics)
Addition
Diffeomorphism
Group action
Scaling (geometry)
Spacetime
Direction (geometry)
Neighbourhood (graph theory)
Embedding
Algebraic structure
Open set
Subset
Force
Field extension
Category of being
Goodness of fit
Manysorted logic
Order (biology)
Energy level
Contrast (vision)
Lie group
Social class
00:00
it has been a time I'm mine on 1 hand and on and on and on and on and and tourists I was told it's discussion reduced starters you also wasn't
00:24
what about to lose not to crowd him the yes so we are there some Arkansas I in the city's so the whole thing and you think it was materials as the number of Christians canals does this project and future I had was something I what would you think directions should devote all what do you think 1 can do reserve something another the awkward explain certain specific aspects 1st of all can you explain this hearing is a even bigger contracts um In the background to fly talk what was once a little more and so would and you know what so the None of animal's sold life sometimes tries to current but the things as generally no free rides for drew the complexity goes up from Verizon's so what what this so what we have seen so far is that it explains how a the equestrian operator in the case are Crossley which I would say referred to as a homogeneous Case tool Sun the moralize space which it not a manifold doesn't stand for many foreign all before that can happen in certain situations but to something which is a little bit wide audiences because then the person was manifolds was rates which globally saw fit together and that allows you to do the integration searing over laws to try relations so that that is what it can do on this level so then so so then you want to produce numbers all of this then you have to do is orientation so so Yukon takes a shortcut that I just look at 0 dimensions faces an accountant plus minus 1 because you destroyed the structure already by using this rated the Branch summit for you dealing with rational numbers .period having rational rates so there's no way 2 1 on orientation questions and so so then you start helping orientation and then you see the direction of morphisms there because I thought did not so I did not not that I did not all of the market forms I did not order the positive ponchos and neither did not all the narrative punch of the new look at the ends of the labor this is given to the give this additional structure so then we also heard about bait that orbits which so so it it turns out that then you can go to a covering so basically you take that solution space that you haven't you go to a covering their use take each object but with different numbering of the mark of the mark ponds and this different numbering top and bottom and also introducing that what you have and the intermediate phases namely that he had this matching wasn't totally markets up to so you put on top and
03:56
bottom additional markers and then you put randomly on the set of prior with a niche player all reported .period so the market and talk should be viewed takes the associated the the frequent park on this issue right along cross 0 process 1 and 0 so that 1 of his era but you you hit that mark .period than the benefits multiple covered time of K possibilities to do so top and bottom so there so then at this point you can take the old bourgeois could and that some ideas about orienting this wrote this but moralize spares technically speaking was lost the letter because no 1 really knew what it was so struck so so there already and those that are in there but but when you look at it in the proves so all the right things and used can take that you can't them to approve it the changes would not find it that London so the source of knowledge depends now the on the so for the grading so if he makes a riot the but grading system ,comma contains a ride definition of pundits then and the right definition should have the property that if it is you go on the contact structured the prerecorded antics of atonement and may so that it can look at that Of course the lira saying minus the identity and that is always plus plusminus this positive under general so the tactics of the Texan heiress return on the continent for and then you can look at this and that you said it's been even altered depending on the sign and the normalization of the condensation then it should be also so that the more to a reduction is precisely that but is 1 of those things not so we can talk about even all comments saying that this is an even convincing the index scores points actually to the size of the things which are a plus many few interchange the so Nazis oppose 7 numbering and the the oriented and then you change the number of people puncture if this boss belong to or what ,comma saying this ordered sign they have to change off on tuition so here you wouldn't knowledgeable about how 1 stuff so so starting point is you want to of course to mumble as much as possible order leave so so you want ozone minimalistic approach so you take you take a patriotic there's none under general then J and order data fixed then we know that there is nothing TalkTalk said a joint operator sitting over so yeah so you you trivializes somewhere but you can think of doing the financing so just put abstractly the captain there is a course treatment of operator which has precisely the told the so you choose an orientation of this 6 to do this for all paralegal they also this up onehalf of sort on 1 disk cylindrical in heaven is so so this is a chance to make consortium so you have this as a bureaucrat operator there the really arise on concerns about that and you have determined by bondage was on to Arkansas then of course they can do the opposite .period top kept here so suddenly let me ride like this so there so there are all united because I made choices so that all the chances and ignorant so I put this caps on there of course no can do during construction and get a complex of issues on which you can actually destroy everything you have cordially month of operated but a growing now coach of month of operators can always be Bahama talked to complex the operator and that the canonical orientation of commerical intention so so this saying has a economic orientation which is the complex on taxation so now we only have to make 1 shot you know you have to make some conventions so you just say for call this law can norms and this not class because positive ones and miners in order to have only to think about and it all you want to write them so let's say I tried this 1st take deep debts of a plot turns up dead of miners goes into debt of groups but this has a complex orientation and this is oriented this induces more so so in the Littlejohn is naturally isomorphic of gotten at 3 other mopping given the change all about then you might get different sides talk so so so so 1 important thing is whenever you see an orientation it's naturally and if they don't use a convention actually the Alexis Ingle started the the high although fury of orientations so so you can have many natural conventions so it's good to know which 1 you to so I let me set out to write his 1st and only 1 of the things that we have this month nobody's abstractly just have to know I have to just trivializes this year agenda that Napster cooperated in the latter half half of the year and thinking she said it was I love and then the here and in the absence of visiting musicians there together with some identification here which affects the currently there in the form of then and now you can only and that was so so now here this for the use of France so I put my so let's make his red so I do my abstractly red stuff in here and then I do my abstractly oriented all this stuff in here In order to justice was so but now not you see suppose my punches well that number now is the what should be my conventions because I have 2 related visited terminal this determine of this determine of this town of this this this this and this and also has a complex on so in principle would say this all should do the same than here it was not so you've got to stuff in it's less complex is a sort of discretionary Monday on things problems but how you do this so so you have to come up with a convention In which ordered to write my determinants 1st long lines in which all of the top so so the easiest array of course is infections upon just all if this is part 1 2 3 4 and is inspirational positive function if they want to and then later convention in the sewer and this is why I have to go to the cover so this will be my moralize space in the covering the number so now I make just a convention for
12:40
the 1st strike down the law 1 so so I ride .period and which has numbered 1 so the tournament the 1 tender terms plans until the last tournament for the last time was determined of the I had canceled her facilities or this isn't just have to get the will of the people to put the surface from the so OK this then I could ride here determinant of the coldest Avenue that's that operators His year so I would there and then you could write down the determinants of their 1 minus that will be for example this 1 than that this 1 and so on and then if you grew wonders this goes to the determinant of the blue and this has a natural intelligence this have orientations this several orientations this that but has said I could also but put all this garbage here most of the nigger different sizes so so he protection so if you write this and then you haven't been doing but is a growing then you then this guy's here matching sort results guys and you just have to figure out which although the each and that is 1 of signs from also your Asian businesses have tomography College signs change but it's so now well this only works nicely if there's no better so you have to otherwise so that the US and the UK so so that say the basics of orientation but there are still some issues With this summer this doesn't to operate and so so you Saugus almost half of the draw on if you have what is so the menu this trivialization invited laughter save our start trivializing sources of forms also clearly doesn't want mark and if I have but if I have another operator which is a Richard is associated to bed so bad all that is what is that even interim off those simple which has an odd number of iron value was visiting hours now has an odd number of eigenvalues between minus 1 and 0 so that if you if you take a trivialization for the for the next pneumoconiosis endusers uphold signature forgot to get the get a different so they do this but it turns out that if you if you use any other all but can you can trivialize starting from any Marco you have you always get a consistent on so the so so the result of business but but the basic features as but then carried out that certain ambiguities to encounter but they're only kill you In the case of that all of United's of battle what you saw once once we have this we can do the following we can actually and everything that's on top of already has good has no bad always on top and no bad always on but terms solve the moral aspect and even to the broken ones went even if the if the result you can do it 1st for buildings of hide 1 and then this orientation actually can be moved to the boundary but in the boundary what will happen is that you might have a bad all of its occurring in the bond but but the horses but then that is a broken saying but the whole thing it said the moralists spaces oriented consistently was in tears what so that's of course not good are so so here we have all everything oriented than the broken saying service oriented but in order to have all of this Europe the should have compared his orientations of the peace no like if you do more serious than each the great in line the most different ones should have more information than the space industry has no intention and then you figure out what the science all right and left things were so here they can break will the universe things don't have warranted however the bad but was excluded vote even turrets of battle so assume you have a bad already the battle is the 2nd in turnover battle zones but it breaks apart the then according existing you just moved asserts in particle of a covered the new move 0 1 of the markers 180 degrees you get a new object it's a new object but now the boundary on intention changes and you can do this together like this and it continues so it's like Billy so that something like that so you you have ends in this picture there's something runs out in this would be connected was that all about and then some of the things continue and this they always appeared parents the Council so if you disregard them consistently in the theory that it does not destroy description from 0 because just said it gets stuck on the boundary doesn't actually stock it's just like a tunnel leading to a different place continues so that there is visible all so they are under .period they just get stuck on the border but which I could think of that there's something else from the other side you would continue questions so it something I wanted you can certainly see it is the policy actually there no allowed contact you wardens although it wasn't 1 of the reasons that I want to know so answers questions enjoyed keeps track of you can make the decision remarks Arkansas so I did what I described today was making generic cultivation this the situation which would allow me to define the boundary operator along these lines so then of course if I made a different perturbation I get another boundary operator and he would be interested in the 4 times all mortgages so we would actually have to study the homotopy from 1 to the other which gives the structure so so that was actually about possible endings for this cause but at Mugello optimistic so didn't have so this so that so this has been locked out then of course all what all assessments to be done this year you want another 1 0 mortgagee its independence which obviously have acted to consider to a different ,comma top and after study on the topic our but I think the over the amount of difficulty so someone of difficulty for the council voted today was actually surprisingly let foreign was
21:23
basically went after to checks things but is that it was also shot the beat the transfers attitude for Homer topping between true such perturbations is a little bit more involved but but it's also the number of things 1 has to point our you cannot achieve general positions on the grid some argues that so there's a certain 1 discussion has to bolster but I think the whole model reporter was quite similar and the next thing is you want to have covered dozens in the cupboard isn't would be also category like this have positive and negative and and then tool of such categories are operating from the right and the left by concatenation but they the I'm not visits to the analysis completed the same In that case and then France's volatility I think it is
22:12
less than what they have seen this is not the body could have seen if I would have been a different selected so that it would be more on the level of former top With such contributions so so it gets a little bit more complicated than today but not much so so that so that that is this picture than you do this on television business and then you do so I think you can basically do whatever they promised in India's of cheap of the French reporters were handled means there is someone who is going be like me yeah but it's so you're you're more interested in time from the delicate occured goes on show and given the experience of amounted that at the time and was coming but but is so the sodium years of the state so there be 2 research papers 1 constructs upon for fall the Qataris current it also has a list they should used for a number of techniques and to build such spaces for example 1 of 1 of the criticisms I can have about the analysis this is the user doing constructions is that you have to do it during construction at once is known locally because of this we have to set up the sting venue you do the drawings and the note
23:40
that this saying size very long solidified them introductions it's unnatural to and then you have to do is there's a lot of things you have to do and it's usually use along and also tried water actually writing it down unless the referee forces so they then then a different versions of shortcuts by saying that a slight modification an appointed and other people rather than you look at section of the also points to something so that's so that is these are in the nature of 6 so what do do where they're coming up with it's something like this you want to introduce works under the notion of a prefect series that means if I would know that I have to say so if I would know for some reason because somebody at some point proved that you have a question operate on something results boundary condition satisfies some that's cannot be frightened because on a bonnet many if any but but that I can put its stamp on it it's prefrontal then if I have some other situations and this is preferred someone and the Council identified things I could say this is preferred by then if I could take this stuff and glued it onto this 1 and there's no free anymore I could conclude its fragile on the London so the pictures that is actually drawings so if you will studies seems like best than you could say there are sort of regions of identification it could represent this body like this so this seemed like a gaffe like this and the other like a golf like this they can identify this year so it could draw this picture and there is no friend it's it's further and if you have so you could think of pictures like this each of us some piece of money analysis of of freedom and as for example going doing you know of the them operate nearby so if I understand this and I understand that you won't of cultural then if I take ,comma here in a candidate to this can then defined this then I have already a serious flaws the Minnesota no you so so then once you have this actually you can build a library of preferred begin the library but if if if if present proves a new piece of analysis and it is it is a after certain specifications you can't plug that into the machinery works so you can start recited so for example the start of the season and somebody does this year this as a piece of our candidacy of course remind itself the gradient flow and proves its professional and then you can plug that into the game he was so sudden construction appears as morphism will center the news the only thing we are seeing but not satisfactorily sold so far on the level of of the operators had called but it takes a while but I suppose the company would that there should be as pretty of missing after 2 years so basically preference of is something that you can do something to it that it's for plus plus some small so this is so this is a positive operate on this this is professional because I can't grew was something because for example I can put Lagoas rebounded conditions on the bomb and then I suppose we have something here and also for them because I can't put a lot and then come to the view of building is so rich and as such is something like that so this soul Soviet missile suppose this area looks pretty much like this area the than then you want to switch from brewing this part together was that Paul and this part together was that Paul and that would be a product of 2 for the operators and you would have proved that if this is for the result was this than this year to glue together together with his business wonders aloud what Obama condition you and your promise for a talk so therefore that is so this is actually known on the on the linear never so the excision former line on Tuesday and it's sort but I think there's a there's a nominal version of finally getting to but himself and so that would be 1 thing for the Freedom Series and the other is extra constructing spaces of small thinks so but construct 24 so I can construct the 24 for the peas .period fall for this piece and then that the sings overlap in this Honda's pieces is a 5 apart from the so there's a lot of fury putting this things so at the end of the asserted the end of the day there's I'm not there's still some work to be done but if you look so you look at the problem you have to develop sense an ideal but also the moving parts and then you can put this into smaller pieces but fiber products and you get the get the uniform of the look at the cost among the chopper into pieces if that each identifiable pieces prefrontal areas in your library and that for the Commission said the it would be the decision was once the last .period only the versatility of rice and the using land rights bill because and I think 1 of the advantages of the furious actually that is that it can be chopped into small pieces I think that that is actually very useful future and so on and so on then then the pieces of I'm Not since you only in our principles but if you have such a situation and you're a new gender just covered by open sets and for each open said you could say the restrictions preferred form and their ,comma than those seen as to be threatened if that would go in this direction then there no earlier than 1 small patches Armenia may always assume the images or 2 and so there's some and you lose some estimates for something on small disco into some are and you have to prove something about this and then it gets even if this has proved that it is a stamp prefer advantages of this catalog of properties and if you have another 1 on this can start doing the things to be really used machinery and Wallace and I will have to understand what the hell you will you have you have to all the other he said well I think the technicalities on this microscopic level usually rather seminars like growing knowledge .period periodic orbits so I think it's good and in life to see at once what entrances but then I think things are quite similar but I think on the level of consumer such a technology you know I would definitely look a little bit into the machinery of men like it ,comma opened the wooden serious delivered what is in the I'm a bit but I want to know too much about electronics but want and I want some boards and then so I assume that then letter some examples of the Bureau of this ,comma Britain's thing the resulting insane up which is sovereign not in this language is written brother down to us in the middle of this powerful technologies it has a certain length but at different addresses all the issues so you see what the problems are hard address and then you get a solution so so it's all good he had once sold solely a few now but of the next thing which comes on just
31:53
doesn't take any president Silva just use that technology constructive saying so to papers this is what I just said about constructions basis and then then you start precisely where my election starts have this categories as new structures on this and I just talk about the leader of the IRA's runs for the perturbations in whole or interrogations and endusers and show what kind of injuring 2 can do is just so just deliver is of 2 parts we what but it's also interesting to note that from this level of generality you see that there should be other things you should be able to do because on this category for example how I have a he differential forms an example of this other the pullbacks by evaporation once have this mark .period numbered that can put back so I have to take a currency of the category you by adding this number and so it is also has structure so that you start pulling back differential forms Form so the concept manifold or you have net into the Lynn Mumford space before forgetting the map since throwing away the unstable demand components and you can put that differential forms from the delay the stepped all of the agreements there will be examples of differential forms on the category and then using those constructors Molterer fall as if to but you could also think Well why don't I just takes the deadon on complex on the category and integrate now this forms will have less symmetry properties because when a predicted not 20 other forms and all sort of went out the number the more poisoned put respect different although I just have signed changes can't predict an integral but but this year and have a few others but it could very well be that this differential forms at the plenty of them this discover something of the topology of the spaces of the underlying altered states and that are more than this forms the use of force there's also lot it's as if you were a good view point that 1 should separate all the I'm not it was then you have seen was struck in the category and then it was just a lot of general procedures on that level that produces use some numbers at the end and then somewhat SunAmerica object larger bottle object and then you start thinking about how can represent the stateowned some others aboard fashion and from that perspective contents of moral duty and cylindrical and saw a possibilities you haven't been but it also consensus to do but but of course that wrote this paper is really was the view sort of brainwashed by life here so so like there was frosty really amendment but monstrosity was there everybody who saw prominent should be ferocity have immunity the new right insisting what to do precisely what has this idea and taken by think 1 should weather should you would too far from today's perspective into can't separate all out there might be a lot of other knowledge about things you can do sold through then you can think about other structure from the categories so I have to say it was in the civilization of sponsor Linus who could have maybe categories which have long since coming off the the kangaroo different categories in different ways from hundreds of supplies and whiners discriminatory configurations of categories to nuisances I'm pretty sure there was a time showed application for this country there like the we can only do so I'm just going used sold works no more so I think I may I say immigrants and there's no analytical obstructions for not doing more study underwater boundary conditions and and put them into frames and paint and and so excited this kind of structure to cling different ends scandal at all different places and so so I think there are a lot of different structures on the level of categories you can think of and there might be realizations as this kind of 1 or more a problem also I think a lot of Martin's inflected geometry get by action artificially created a new type of operators and things just 2 true tube to prove some geometric effect by counting some real moral aspects some so if you see some of that that you can put the sings smaller pieces so it becomes more like a puzzle that puts things together and so on so so it is a language use of freedom I don't and I haven't talked much about it except carry out this as if to but but but as saying the optimists and so I would think that it should be some kind of like domino pieces to complete them to go and abroad on the news the substance and then you get someone was to represent something when itself look at the present some genetic effect and in the constructions will be much faster than the firm's such publication of going through what kinds of things invited us in and many of his preferred answering there are just as Western Europe and so I would like to say is that it is the end this year probably in April saying only that it was the only solution is that all the have use that will be the 1 usually hears about the security of the company of the dead and I don't expect that those who want more the reality is that we have a right to do all the stuff I of which you can do it but you have to do is the breeze but also you so what part of the government was later in the day of the year absolutely and what was hard to that we have also if you have a lot of work to do in the east to the site of generation that builds the generations have grown up with this story lines and so would have entitled particle has happened in the world series several years to write the parts the beginnings of areas that have been used to localize the analysis that was the need to use the word rapidly in the 2nd period of my State of the whole thing is that this will week subject of the day I have a problem has been solved all this small service standard analysis of what happened was that the rating noting the company has studied this service was to be announced that it will have to ask you Paul all 3 problems that the size of results and school we have to use there is a lot of work to be done relatively harmless you know I want to see the role the pot and so they can get out of reach and fall out here we have a lot of things the way you will vary by region and communications in the area had been seized work not the houses were selected because were women In the past the election race is if you want to think about how we were worried about the direction that is accessible to Yale grows were freed people will soon be seeing that in 1 instance in my view is that of a variable interest region rate rise let us out this world .period have guys think it is a clean way to think about it even if you don't like them like the most and come up with owners of the differently if you're not so so is it's like going to put things together and who died there should be a ceiling market don't believe there is at least 2
40:06
can write it down and see what the results should be in the approved by different but no 1 but the the question is it really but then you are have to make a decision that want to use a different message just learn the language especially his you want to do it authorities said I know what I would call a researchers beginning what is the sum of city the a recent of you know you will need to find the right you have to on framework going soft on increases in essence that the country of origin is regarded as 1 of the trial of the workers so that we can do something similar in which you want you write so you find the right the right for a policyholders will live below parts that have already been covered bridge and was being the 1st the constructed in relation to the meeting of reality was right and that several of you to the the the decision during his final years his cellular or a New Year's address to the nation said that while he would have really understand here here I you have all this money others interviewed the money here sold look at the Aga Rosen so there are no things which if I put Tokyo like this than this piece of perturbation .period appeals or up everywhere because when I goal to have new phase it the appears product and later being used and so on so for that might in some problems the true generosity issue and yet doesn't for the deal for the boundary operator but you could imagine that that there's certain things that have so many conditions that run into different dogs so that it could be that is the and then you would have to allow for certain mistakes still form that you have to allow for more general perturbations and then you get some kind of higher structure which keep track of of the lack of being actually able to be keeping all the structures being transverse I have decided that instructions the mistake you have to make to get its cereal on the level it is so much chance was honored the Newton was something might still we need to break symmetry but there are different ways to break the symmetry and you keep track of the different ways to break the symmetry and then go to the next level and that is not about object answer he also I think you and you know when you really want to intersect the diagonal walked through news breaks a signature also like like they're on the way down on commodities on the level of the chord changes is a product of the singular isn't we have to perturbing keep track so so so I don't know here in this way but I could imagine in effect just put so much structures on this I cannot just achieved standard regularity so so I have to break to break that I have to give up some of the structure but that but I think in general and there's not because it could be in the back on knowledge about series All in which way you can give up this structure and then keep track of this ability where Albany Avenue mathematical object the government of India the 2 were just so so in most parts then so so he or in this case Of course 1 part is the Gloria carrier all so I have to understand that picture and young mothers including so even most box well then in the neighborhood this a mothballed manifold you would least describes analysis you so out depends not only 1 to give US sales some people like to put a multifunction theirs but it could also what does have this and swanning around and and individual logic losses but 1 edition of the hands and there after the sudden loss of our sponsors so so it depends how you want approach but but they the end result is the Sony building so far this year becomes a little black box which looks like this and then you have something as which I worked on the duallisted and I something here and this is the others so she have to build a black box you have urged starts overrides himself and then Jordan into the the explained undistorted but reports presence and it was unclear what it is up to and from Sonia there is a section from the reader yes so what is sold so they but is so they Benjamin so when we decided to deal with the set off on the tree 1 to the freedom said not 1 that you want to work then Our decision was on the domains in every room we want to have local construction they look at you could imagine if you have a space all MEPs from some offices which also but can be glued to could think of putting a structure on which being obtained by solving some people and on the it would not be of local nature so what we want to have Is that that there is a locality to this if I know the local properties around the nodes everywhere I can recover from this structure of suspects that is what makes it possible that the controlled into pieces this be that I just always point was then New Year's and also because of course grew and we have to do is a problem that get different so so then then you when you want to have this function spaces of this topological spaces has very local property then then there seems to be no born space or set for modeling even if I would only have endured notes and Albright this is the best that vote a space of maps the final on an order you want to discover neighborhood they drew the cylinders can as far as I I can see cannot be modeled on and on the considered so so therefore so that so so that the vote in all sold by exploring this in there was a deal for putting this aren't arguing and then you you got some kind of a message substances 1 which on which you could define the chart the coverage of the later turned out to be as a redirect and the redirect then turned out for the season was part of the reason was not for the news was for this city ,comma grows so the had actually those most moral for space but it was actually reach the respectable less 1 after another if there if there if there would be a small snap onto this redirect this reader what it should be borne of something for but has actually bring dimension so that that is so so in the classical sealed the victory Oracle Thomas lost here the continuity of the deficit has lost a net must published work but the child already learned later from 1 that it changes so so he proved that if you if you have In about status and was from the said would serve and I ask
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for composed of because of the images about that is about
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ourselves so that is what improved and what we have done all this also have to get this off sold because the realist was there was a retraction libelous differential will fall for this is Congress but then of course natural question what is what is is actually if its classification in the office and also feel that it has to be 1 of what do but in it for the city retractions in general it is not the case with the view that there is a lot of space but then and and then of course once you know this is the 5th was retractions for for any notion of the French abilities many of the chain rule so that it can do that then you can be a theory based on the 2nd Texas images of attractions models the market should even different forms of the French ability is what we would like to try they also so they had to vote once you have so Wisconsin was born oddlooking political group always molds and so on and it turned out to be role as a useful and I think I made that remark before there that may be even classical differential geometry should be made this redirects because a lot of stuff this is simple so if so for example when I have so nobody would really like to define this as a supplement for classically because the positions so if it is to take a differential geometry but that Jeffrey onetoone to say that the something but this year a subset of the local richer his taunt you have retraction so the better definition of the To say the submanifolds is 1 which is a is the image of a classical moves from and then this is all fine and it has attendance basis since it's a metaphor structure the on thing what current Syngenta in highdimensional you cannot say that the that the boundaries in some form because my peers a what however if you know that on the boundary the tangent space is a nice position to the hold that it has structure so you can't say if comfort like this all the highdimensional versions and in all the inducing uses many for born recalled so redirects actually was for been on this and and then if you want a good many of maps that was really fast this was supposed that's a bonus is not all commodity groups of 1 a new experience of maps into some of In the end the only thing after notice the fastest most helpful in into our area it's a visit is affects and of course against Lewis composition into the the 2 classes of so so now support so I claim so and this is fun told me also to every index and you haven't heard about spares and forcing was not for life into RAM again inducing was operated between those who expects this thing has a canonical extension too many forms so I wanted to the best dozens of what some would say women there so soon supposedly many for the Kenyan better than a foot into a sufficiently large and To some other competent default if not you have to properly but it's a competent formed he has not
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is a metaphor take it to the neighborhood here Natura neighborhoods and then you there's a retraction from this neighborhood into it said those images of metaphor so no this 1 by restricting the smaller you can extend to a map so by cutting its near the bottom in sudden so by restricted slightly smaller neighbor you may assume this is a distraction from map From now on the enjoying so soon now you take the open subset of this thing you're busy image lies in the small neighborhood the soul takes a subset so take all you wish and at age 3 of this in there is the image of an lies in their neighborhood this is an open subset of the of the space In defiance all traders by all 3 those of you who is our composers you but since the restrictions he was never this was met and it has a property Australia drew 0 and 1 of them but images all maps which goal In 2 0 end so therefore it is at least a point for an employee for some it's so now take a different embedding into some of them then the only 2 embeddings diffeomorphisms and extended in both directions was was so you see that the twist things are actually as soon so therefore this the upstart space of all MEPs into M which together with new betting on all 3 has a very defined imported from structure and then you think a little bit and is actually assume any false structure that wasn't taken we try to contain a spaces in the morning so I think the notion another action this case it's in classical on each level so so so in this case actually on H Street it's up to you I don't even have a scale he discusses his moves so this is most suitable spaces like it's actually it's 8 years he's actually a classical it's as important for structure for classical retraction and by contrast series it's essentially that there about supplement for so you say you want something like so that's the 1 thing I said If you do it might depend on the embedding should take a different embedding then of course it embedded seized by this man but if you more often than for each of them to take a take a somewhat extension was no broadened his head on the whole of the
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nominal nonemergency was not it's not so so he'll this is just classical solutions not so yeah so sold so what this proves is that the that the space of me is that there is a subset of From s into Ryan of classmates 3 image in em is the classical retract for the Class a different structure and therefore the parents they have a meaningful and this definition doesn't depend on the bed you just look at the next interim which together with embedding of his property and if you take a different indicating that their destructive additional fixes economy structure but I am so sorry sort the case in the middle child fugitive wanted more about if influence on the social disapproval structure not so parents or attractions good stuff 3 minutes closing the into fuel on the scientific community organizers also in the end we should do big applause too the years from it as possible in order to achieve also true all organizers erroneously so Jordan's support then and Jolyn ends tours To Jones you'll do a lot of work jewels movement and had to go to parents so that plotters the few mn