## Moduli Problems in Symplectic Geometry - Discussion with Bottman

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 Title Moduli Problems in Symplectic Geometry - Discussion with Bottman Title of Series 2015 Summer School on Moduli Problems in Symplectic Geometry 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 10.5446/16314 (DOI) Publisher Release Date 2015 Language English

 Subject Area Mathematics
Area Trail Hecke operator Cartesian coordinate system Mereology Dimensional analysis Subset Number Connected space Plane (geometry) Right angle Near-ring Resultant
Functional (mathematics) Octahedron Multiplication sign Projective plane Infinity Incidence algebra Substitute good Number Glattheit <Mathematik> Energy level Right angle Series (mathematics) Funktionenalgebra Homöomorphismus Spacetime Condition number
Functional (mathematics) Surface Connectivity (graph theory) Projective plane Cartesian coordinate system Subset Fraction (mathematics) Glattheit <Mathematik> Normal (geometry) Right angle Analytic continuation Fiber (mathematics) Homöomorphismus Spacetime
Point (geometry) Different (Kate Ryan album) Infinity Condition number Arc (geometry)
Difference quotient Point (geometry) Functional (mathematics) Beta function Divisor Differential (mechanical device) Multiplication sign Connectivity (graph theory) Range (statistics) Student's t-test Mereology Product (business) Force Many-sorted logic Term (mathematics) Analogy Negative number Modulform Energy level Stochastic kernel estimation Series (mathematics) Position operator Sigma-algebra Forcing (mathematics) Projective plane Expression Infinity Limit (category theory) Approximation Shooting method Numeral (linguistics) Glattheit <Mathematik> Right angle Musical ensemble
Group action State of matter Interior (topology) Multiplication sign Correspondence (mathematics) 1 (number) Set (mathematics) Parameter (computer programming) Mereology Dimensional analysis Many-sorted logic Negative number Partial derivative Potenz <Mathematik> Process (computing) Moment (mathematics) Curve Infinity Maxima and minima Price index Proof theory Order (biology) Linearization Normal (geometry) Right angle Spacetime Point (geometry) Functional (mathematics) Wage labour Heat transfer Theory Force Frequency Profil (magazine) Well-formed formula Operator (mathematics) Interface (chemistry) Energy level Boundary value problem Modulform Analytic continuation Condition number Scaling (geometry) Algebraic structure Limit (category theory) Cartesian coordinate system Voting Network topology Universe (mathematics) Family Funktionenalgebra Local ring Near-ring
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
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
the function after you a got here there you have to a world record for bad stations to use thank you
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 x-axis is little tea and done well so there the fiber living above any particular positive team in the fixed-point said and it's going to be the some space and 8 0 stand by it and the fiber over
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
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
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
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
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
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
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