Moduli Problems in Symplectic Geometry  Discussion with Bottman
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Title 
Moduli Problems in Symplectic Geometry  Discussion with Bottman

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
Area
Trail
Hecke operator
Cartesian coordinate system
Mereology
Connected space
Subset
Number
Plane (geometry)
Lecture/Conference
Hausdorff dimension
Right angle
Resultant
Nearring
03:07
Series (mathematics)
Spacetime
Octahedron
Multiplication sign
Projective plane
Infinity
Incidence algebra
Functional (mathematics)
Substitute good
Number
Glatte Funktion
Lecture/Conference
Energy level
Right angle
Function space
Homöomorphismus
Condition number
08:53
Spacetime
Surface
Connectivity (graph theory)
Projective plane
Cartesian coordinate system
Functional (mathematics)
Subset
Fraction (mathematics)
Glatte Funktion
Lecture/Conference
Normal (geometry)
Right angle
Analytic continuation
Fiber (mathematics)
Homöomorphismus
11:21
Point (geometry)
Lecture/Conference
Meeting/Interview
Infinity
Subtraction
Condition number
Arc (geometry)
13:36
Point (geometry)
Difference quotient
Musical ensemble
Beta function
Divisor
Differential (mechanical device)
Connectivity (graph theory)
Multiplication sign
Range (statistics)
Student's ttest
Mereology
Glatte Funktion
Force
Manysorted logic
Meeting/Interview
Lecture/Conference
Term (mathematics)
Analogy
Negative number
Energy level
Stochastic kernel estimation
Position operator
Series (mathematics)
Product (category theory)
Sigmaalgebra
Forcing (mathematics)
Projective plane
Expression
Infinity
Limit (category theory)
Functional (mathematics)
Bilinear form
Approximation
Shooting method
Numeral (linguistics)
Right angle
22:13
Group action
State of matter
Interior (topology)
Multiplication sign
Correspondence (mathematics)
1 (number)
Parameter (computer programming)
Mereology
Maxima and minima
Manysorted logic
Meeting/Interview
Negative number
Partial derivative
Spacetime
Process (computing)
Moment (mathematics)
Curve
Infinity
Price index
Functional (mathematics)
Bilinear form
Proof theory
Hausdorff dimension
Order (biology)
Linearization
Normal (geometry)
Right angle
Point (geometry)
Wage labour
Heat transfer
Theory
Force
Frequency
Lecture/Conference
Profil (magazine)
Wellformed formula
Operator (mathematics)
Interface (chemistry)
Energy level
Boundary value problem
Function space
Analytic continuation
Condition number
Scaling (geometry)
Exponentiation
Algebraic structure
Set (mathematics)
Limit (category theory)
Cartesian coordinate system
Voting
Network topology
Universe (mathematics)
Vertex (graph theory)
Family
Nearring
Local ring
00:03
so all told my name's native Yemeni already uncouple announcements there are solutions to the homework exercises which will be put online I think later today but there are no official officers but all hang out after Michael lodgings lecture to dating is even questions all right so welcome to the pointless discussion the primary purpose is that I suspect that if he hadn't seen bottles before Monday near bewildered about at least part of what's been said you know like some piece of terminology that does make sense like maybe you think What the heck is this partial quadrant for whatever else so I like to take your questions and clarify whatever questions you have so without further ado money turned the floor over to what questions they the 1 you yes yeah I and II out so here's a question for my calls but certain all these anomalous results in the coming year OK area so that this is a track which is locally finite dimensional but where the dimensions vary within connected components is something that Joel alluded to earlier today and this is also a common question number 5 and so the fact is that there exists a retracts which is homeomorphic To the following subset of 2 so this is supposed to be the right half plane union with the XX not included not including wax right so is satisfy a definition of weird I'd like to
03:09
use this as a deal that you think itself it is all all of this is that I so you know this from my find amendment would take the a life and who work is not the installation was and this is exactly what has been from the policy according to the apology depending on the it is not in the country that is not right the policy that could vary with some of this is that of and it's absolutely key so I must have a stake Weissert homeomorphic today right now I don't know what you to think about all it Francis creatures searches would put a different apology which was not always so long all right so the idea of this is that we're going to fix some functions and bond function data it but and dumb for serial number and down In the DeLuca a retraction on function space which is just projection to Baidoa but shifted by asserting that depending on the condition that concerned parties grievances and then 13 less Naples 0 just consider the 0 back and as the funny thing is going on here is that and so team is living in the x axis and as you come in from the right so teens getting smaller and smaller and heat to the 1 over 2 years chasing after infinity sir projecting onto this a series of bomb substitution after Infiniti and the interesting thing about skills smoothness here is that these maps behave smoothly at 0 so something funny happens is 0 but we're going to get it deals with that all of the a lot of work already done but it's going to go from not less right so this is that some space in L 2 Ocanaz stand by that shifted the function the last thing I would like to think I could play write that down right a precise definition of that so let's say said the tubular steel space which is what the level is HK Celtic k functions from Oct R. I. where we've chosen an increased incidence of Delta's starting from 0 the and can now out retraction the candidate retraction little are those from and seek to our time Missouri art and eat our Tennessee and as defined by sending the pair he Little 2 the projection onto the team 20 is greater than 0 all right in the 2nd 2nd Beatty means the shipped about function as the head of I you know that I don't want my I'm sorry I should call
08:37
the function after you a got here there you have to a world record for bad stations to use thank you
08:58
yet this is supposed to be to poorer ,comma fell to at that of any other objections great and so I think it's clear that the image of little are Is is the homeomorphic or maybe don't leave the continuous news but it's plausibly homeomorphic the subset I drew above where the xaxis is little tea and done well so there the fiber living above any particular positive team in the fixedpoint said and it's going to be the some space and 8 0 stand by it and the fiber over
09:47
anything not positive is just the story about the right with the world right you mean right here yeah you go from the fractions work yes I accidentally written only the 2nd component of the surface was to be the ,comma the projection of passed on to the some based can buy OK OK it's the yesterday that alright owner before and we want to proving some amount of smoothness of and confusions about she was obvious why doesn't really matter it's about function with but With H is norm equal to 1 while we just say at the end of day and the
11:23
infinite this is I think probably if if it was positive and also made this it's 0 ,comma Delta 0 and would probably work the question is if if Delta 0 were positive and not 0 would still work I don't think the what the move nobody here believes that by the end of the day I was not the case with the he you should not think of all anyway it really doesn't matter because as
12:25
an example so as to say that Delta 0 0 no talking and let me just before I proved that this thing is somewhat smoother Is everyone happy about this claim that the image of little are as is set that I drew an arc to OK I'll take your word for it all right and so I'm not going to prove that Little Arias like SC infinity anywhere because are everywhere because I'll take forever but I can at least show that the difference pushing condition is satisfied for proving as 1 but at points of the 4th 0 ,comma the people of Hong Kong old this is how you know is that is all these things call on you what this is and this is actually an important example because it's not exactly the
13:36
same but it's related to the possible but because in prosecuting while you're not projecting onto some lung function but your multiplying by the bomb functions where or something shooting off to infinity and you need smoothness of the retraction that corresponds to that so the same sort of you know that the same reason that this is an assistant is at least part of 1 the attractions which is projection of the kernel of yeah I guess the size analogous but the idea is that in projection to minus glowing when aid is equal to 0 years doing
14:29
nothing if projected the kernel of my schooling but I would have to think about whether it is the same but there's something drastic happening at musicals are Monteagle not also called I really want to do the same thing all this said OK great so it turns out to be true that the are is Casi infinity and therefore a skills with the tracked with the differential like cell OK so the case that tea is positive missile Sigma ,comma negative signal 1 over peace acts of 1 of the team than the quantity at times .period prime 1 at times cleaner and then plus block the data needed yes no but I claim that it's trivial 13 negative but it's not trivial I think that the most interesting things happening at the polls 0 OK but so anyway this looks horrible but it turns out that it's not easy some of candidate but anyway I'm not going to use this because I'm only going to be looking at what happens near Earth act equals 0 so it's just show that if we're right there are is equal to T ,comma for you to yet so he is just the 2nd component of our the component nothing to eat then he is scarcely 1 at a point in the form of 0 ,comma so let's show that difference quotient that statement indeed goes to 0 yes so different question is uninteresting when sigma is negative because an assistant with 0 what you want to prove is that if you look at the limit as signal decreases to 0 and the goes to 0 each 1 fell 2 1 and then I think you're looking at it is after signal signal plus Beta signal the signal and monitoring the 2 0 but by statement from each 1 of the wonders of Murphy he claimed adjustments equal to 0 and so and this thing is it's just the team evaluated that Sigma ,comma T plus at and then the other 2 terms range and then will say that this is less than equal to the limit of falling 2 1 that what world on it's because the differential appears in here is the differential at but he is equal to 0 the company here in the interest of you and you yes we have to get that candidates for the differential but I claim that if you're if you pretend to be a popular student who believes that everything is true if you do that not cater series approximation then this candidate will easily popped out its write down what the differential is 1st approximate OK if you're tired of this example I promise of an almost OK so we need to show that this limit is equal to 0 and and let's just proved this for 1 of these factors and the other 1 is really similar problems so let's look at this guy here so the numerator is the product of Evan Beta Sigma and Beta Sigma taking major enormity Suriname Beta Sigma 1 so we just get the acid value of this inner product but but because the Suriname of basin is 1 that's bound by that means that this expression here is bounded by the it's Suriname of now Over absent value of but then right we assumed that for answer and West 8 0 I mean it's the only thing started the only thing about the product is that Beta Sigma is a function fondness he denied negative 1 Sigma ,comma beginning of 1 of Singapore's yes OK and now we can down this by some exponential yes at Preston the only thing only take pizza animals have on this in turn has led to the this friends and now using the fact that that EPA's on each 1 level and therefore on the 1 level in their force in each 1 ,comma Delta so
22:16
you can down this fight even negative Delta 1 times the 2 1 over overseeing minus 1 after each 1 fell to 1 Over absent values Sigma and the other this is just some constant were not you know this is fixed before we take the limit but In this quantity goes to 0 faster than segment therefore on his way the but yeah so what I was saying is that Beta Sigma is supported on this interval just this 1 year if you write down the definition of each 1 ,comma Delta 1 and it's just immediately because this norm is that each 1 normal asked X negative Delta 1 modern times and soviet it is right on condition of this I Sobel of space the spot what is the origin such a wooden right down but in order to really need is different Bush to go 0 for 10 in the 1st level and therefore this norms OK but added that this sort of raises the importance of the fact that we chose this particular gluing profile that or functions were shooting off to infinity like X 1 of the surviving going after finding much slower than that this limit might not have ended up being sought investors OK for this moment the if you choose some Rivaldo Tiffany much faster than exponential would there ever be a problem it was should also agreed that the the costs of the war only them all of the people of the world the this like Bloomberg is the mother of all 1 all the world the of also 2 move of the bodies of 2 OK the next question actually the reiterates the government also yeah you know I don't think anything sensible about why this retraction actually be homeomorphic the the substance of our 2 but if you can that I want to play and the head of what's called well you know by definition it's inherited from the topology of the ambiance go on space 0 level that but I know that's the sort of thing you had in mind yes of presumably it's it's some
25:50
fairly easy exercise to show this is on the more expensive support all right I don't know what to was going on Oh you're asking you're asking Could you get and so 1 of the wonders of losing his region so "quotation mark we're so that the issue of the prosecutors of the world of the region in which is still on the loose ball and it's all about the intentions of the government and the states image we want to do it but I would also say that it was and this could be due storm it was so the people so that they could be restrictions which they say they want to stay president this is the story of a branch of military the problem now you you the election material or so so they used to call the solution of the problems strong words much of the interest on the part of the world you go you she that was the right corner of the world what is but then you would have a lot of work we it was the 1st unity supporters of patients you as the 0 set of the transfers a mail last and mentioned that this example is that is a good exercise to complete their conscience speaks to this scale retract and in procurement you might wonder what is the dimension of the anticipates the origin told him that as a mystery the on so many other questions OK he knows how to use a lot of people the yield on the what I don't know what was the start construction of England you will the Polish people from all over the world from working with the support of the what was the work of art and the professionally what tools referring to is that this particular retracted and written down as the special structure which is that it's family of linear attractions and there's certain things that are true for and the special kind of retracted recalled skills placings Archer for generals still attracts I I think the and in all applications currently and skills places orders and I think my philosophical point of view correct me if I'm wrong and that is that I like perhaps there truly awful skill retracts like the campus that I don't know but I don't think we should worry too much about them we should worry about what we need for applications and are colorful flexible too flexible enough to encompass them so I think this 1 is relevant because it's really similar to the retracted get coming from is not there was 1 of the things you do that you have for the mysteries of the universe this 1 it's of role is the more that's correct and I guess you can actually see it very easily from From the formula OK I lived there are no questions I can I can say something hopefully useful and if you want to the duration of the the well so OK here's the answer to the dimension that interface for every .period In there with TeliaSonera ,comma 0 it's 1 for every point right it's too so then so you could conjectured that you have fun what is it sunny continuity going down to jumped down so I don't know the introduction of the common not right now OK so the example that due mention where he replaced baited by 1 minus Bader of course the dimensions of the infinite but you can't you can see the it's sort of have the the officer comes continuity with that OK for rights so last chance for questions are also say something for the remaining 5 minutes of action discussion the United States I could probably won't you go beyond that you can tell I think that's the whole of HW wins but I I don't know whether there are any particular examples they have in mind have but what is well I mean I don't know if you include the granting of the year whatever notion of assets you have in mind but I I interpreted as can you construct things they're not modernize pieces of geomorphic curves as revised the that is the 1st in the world ,comma because I want to what's going on with the story of the election because will Christians there was no good Labour's at the moment there is not 1 of the most important of his books all the conventional mortgages made use of it was the only person in in the world there should be no problems whatsoever it has been a success this aggression is yes because theory certainly could be framed in terms of political the Western coping alright so the last thing I wanted to say is I thought that it might be mysterious to some people what the point of partial quadrants so I thought I'd see as a couple words about that so right so what's the point of looking at spaces like never received this it can be but in this instance the question of what we consider part of quadrant is what happens if were trying to construct a much less space of geomorphic curves and 1 construct a local charter near something with no yes boundaries so supporting Lagrangian Florida's for looking at all of a With boundary on Lagrange and we try to construct a local church near something with the boundary notes of something looking like I like and then the local church near a map like this is going to be in the form of 0 ,comma 1 times 4 times and they of space of functions and the answer is that the reason that this is on the right sort of local chart is that we have cost wants nearby maps nearby maps to the sky to include where we smoothed out the nose and smoothing out the node is 1 parameter operation corresponding to you know how much it gets pinched and when the imprimatur which buttons here goes to 0 you get actual nodal so anyway so that's where the partial quadrant definition coming from what is it 1 place comes from and as a closing question the audience if you haven't thought about this question before Monday let me ask you find questions takes but some map with interior nodes something that looks something like that so like this is going to have to generously index just because this is the local chart and when you go all the way to 0 you get an overlap but the question is what should they DeGeneres what's expected the generously index of Annapolis that as I stupid the question is but if you construct a local public .period near map like that and what do you think that generously index of this matter In the we remind you that there are too many printers 1 neatly framed the and 1 angular twisting France for doing that year you can't vote if knew from 1 1 and 2 Bloomberg News the yes so so if you take them up at a local charts but then for this particular chart that agendas Unix corresponds to how much of the printers living in their 0 1 of care 0 and then to get the designers in 6 of a point of Polly politican minimum overall job ones anyway the answer is 0 and the reason that 0 not too is that they're going parameters for the snow In the disk so the distinguished .period at 0 corresponds to Noble maps in the move away from 0 then you're going to know so in fact this is an entry point disks the junta generously index will be 0 also this the like the interests of please don't have yeah we find that there was no longer the case he that is what I want you to think you can is you we need to make some kind of proof you have a particular situation that there would be some other chart where have a lower generous index OK I will stop there