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Symplectic non-squeezing for the cubic NLS on the plane

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Symplectic non-squeezing for the cubic NLS on the plane
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We prove that the flow of the cubic NLS in two dimensions cannot squeeze a ball in L^2 into a cylinder of lesser radius. This is a PDE analogue of Gromov's non-squeezing theorem for an infinite-dimensional Hamiltonian PDE in infinite volume. It is joint work with R. Killip and X. Zhang
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so the time and you have to 2 the latest start by thinking on the organizers would inviting to this program is my 1st time here and I really enjoyed myself so let me start by giving to the of ultimately join the treatment needed to stake out his other than of course I have to tell you about the classical cannot switching result of grumble so starting slowly what is a simplistic manifold so I intact manifold is a man who is the one-dimensional endowed with the simplistic thought so I wondered does this forum meet in order to qualify as a synthetic loneliness would be closed and on the generator that would be good to form the canonical example of assimilating manifold is the following Connecticut it's nothing else but are too powers to end the game show .period disease confined clearly writer Chris takes home of the campaign and we view X medical Texan gathered of fighting and X's including the positions also in particles in ah impede the movement of those particles What if you wanted the position and
momentum over a single particle United States with somebody of any sport accommodated in dealing with the 1 particularly not for particles because enough to ward of impropriety what is the canonical synthetic form is due be won by Georgina 1 past Cleveland which the extent and it's clearly a case of 2 former which is closed in on the generator and anti-Semitic and now this was simplistic form is the 1 that is responsible for Hamilton's equations in their traditional support base of please 120 in the the onto 2 in the quarter and economy then the fuel generated by this Hambletonian is defined as follows only all Dole is equal to the adult above disease stands for the time derivatives as he is a function of time taken various you right now this is the differential of age I can't see any input poverty and whatever you put away now even the Cold what this means of using good the definition of Omega you covered exactly Hamilton's equations my X J cult by the exchange 3 officers he plays their latest album due and PGA billed as mine has been linked and the election of the I don't want him phrases rephrase this canonical examples of his
father was so afraid of this all can regard the on to the whole to end its casino and and we arrive at a point of the is 8th last like the movie writer to the canonical form as follows almost got of the and on the Simitis the imaginative part of the world that between the victim of clinical the incident and in products are seeing the 2nd injuries but that's convention and it is not as much as the majority part of some of the injured bar the budget program now the advantage of rephrasing the canonical examined in this way is that it easily generalizes to all of us in respect the what it simply because they face In this savings and human space age group a producer complex industry endowed with a simplistic form defined by a minor steam engine and part of the border between the inventor and now live in a world that is in each case this is the setting in which the nomination and integration can be seen to be hammered poignant With the underlying synthetic cubist bespeaks being held OK I will tell you more about that soon for now and we just to be treated to the final dimensional sitting in explained to you what the Jamaicans quizzing the result of grumpy "quotation mark Hamilton's equations have structure of paying for the new right here haven't won in filters preserved the simplistic form only and to introduce 1 piece of vocabulary that is they are simply from office and so what is the simplest and most of them in the office of the reservists the synthetic for complete so tolerated mathematically what does that mean it means that if you look at the Hambletonian fueled sometimes teams then the quarterback of the synthetic form provided him a poignant 22 is Curtis contesting connected for a band that is true for all the times to complete it is convenient and money perhaps more relating to write this down in endangered for people he said we agree on so quickly that no 1 we can rephrase this relations like this you have to copy is all and the Hambletonian floor with the stock it takes 1 to 1 of the UK now if you have a
two-dimensional surface as over years we take every single point in the two-dimensional the two-dimensional surface and we free all according to the Hambletonian flow with plenty to what recover is another surface like this in this relation over here there is nothing there are the following the integral awareness of only about equal to the of all McCollum told her that the image of the service areas under the Hambletonian Bloomberg no what is the name well if if the complex dimensions and you equal to 1 then only is a volume for it's after the and on the generator before and can only go up against the morning before the so what we've seen in this case is that have been put on hold the company was preserved area compared for the office of the area off base conduct the unemployment lower apparently now if the complex dimension in the streets would be that the 1 then holding God waged only got in times please avoid as among the generator 24 they now because time is money inflows preserve the simplistic form on a guardian ,comma preserve the volume for the water to cover the use the following theorem often reveal questions that fuel users face this said that this leads us to the following natural question is presentation of volume the only obstruction for the existence of a In case let's write it down question it is resignation warned the only obstruction the existence but office in compliance with these awards by that the the picture if I have to do all this holiday so he is low number 1 and that is the number 2 1 writing it and I know that the volume was low number 1 is equal to the volume of old number 2 does there exist at present little morphism political commodities and between the 2 adults no if so Moser proved that if I have to belongs all the same border with a property that the vertical morphism between them that is they have the same topology then cynical morphism that preserves voice you may now if the Congress answer if the complex dimensions pain is equal to 1 pain and I have to involves all the same volume and his and between them then Moore's that I have a diffeomorphisms that preserves the volume that is the simplistic for made of admission that it is assumed that amorphous so the oil the good news is that an Encino mapping every single faction and William preserving what will wait to see if they receive it isn't what and that there is 1 that preserves the board 1 what on surfaces I would think so the answer in this case is yes creditors due to a more it higher the complex dimension is strictly bigger than wanted the answer is no right and this is almost certain that condemns continue the Minitel you want that OK so I have I country that is this conference to writer and then I will never reach it again so girl here and I didn't what does it say Well I imagine that they have a ball said that the disease .period free upon services employment In the end and imagine that they have a cylinder the OK which 1 write like this seat are people on Fox and held OK so the family that God is going to be the radius of the 7 days positive on fire is going to be the center of a theater complex number and even element of scene which is normal likes to have let go on so what is this assessment was that its Prince cuisine 3 and where the property that the height because the In a product with a minor side of all this is good will to power isn't what does that mean well l determines a substrate sufficient 1 complex dimension when if you want to real dimensions so worldwide over here I've taken VI projected on that sub space in my seat will be needs school even within an hour of this sent over here all right then Parliament and that was in little more than flight pay now or pay a fine all the Balkan currently following takes the ball inside the cylinder then unnecessarily little beat would equal then capital of so in other words it seemed that the morphism cannot squeeze a ball inside the inner cylinder despite the fact that this has finite volume and this has infinite void OK so this is a time when the point where people like to draw the analogy with it camel which can also boast squeeze the eye of the needle in now this is not the few have seen didn't before this is not exactly a traditional .period relations to obtain the traditional formulation you simply take Ellen could be even 1 the 1st 3rd victory in the farm item basis for CIA but because in that case to the Senate that becomes commemorate like this is the point of view seems to begin with the property that makes me want minus the real part of foghorns squared plus the 1991 upon the colorful squared his small then ask whether this is the way from almost enormous typically written however that a relations savage entirely equivalent to 300 Cleveland so how do we see
that as well and because LO is normalized to have length 1 there is a Unitarian maps that that makes you want to help right and unitary maps preserve the inner problem which is that the preserve the simplistic form so that if there is I think that the morphism between the ball and this city and that the composing it with a unitary navigators in the waters of the maps the ball inside this city and the other way now before I move on to the Internet thinking I would like to give you a few examples of squeezing of and they're not going to be counting go almost there and we just serve to better appreciate the hypothesis of grumbles the peso examples of squeezed the bank and that so there is a to marks that I want to make over here the 1st of the mind is that 1 might imagine that the reason why 1 can squeeze the ball inside the thing Sunday's because for some reason the board is very fat in 6 1 P 1 direction paint and however that is not the case and he's examples OK that is not the right way to think about it that's considered the Hammond age to be minus ,comma he won the extremes of people takes 1 then the flow generated by this Hambletonian logic and write on Hamilton's equations if you want the following through on its 0 we used a lot longer than is the gold becomes your mind a strong bond 0 up like disease OK so this is nothing else but rotation practices your In theory explicitly is coarsening of teachers complain the sign of the sign the "quotation mark information about an initial got dumped but 0 OK so why is that good news or bad news opinion on how you want to think about it but if you take the is in the show that pump activity condition that aligning inside under heat source which is very fat in the excellent P 1 direction but seemingly it's the people direction and you through lowered 4th time I will work to toward OK then Muchadehama is now a cylinder which is very flattered but because they the cylinder which is very fat in the export between innovation but that seen in the excellent 1 directions and particularly have known this of fact that it so it if you want the inside of the inner cylinder OK so the reason why 1 cannot squeeze the ball inside it in cylinder has nothing to do with the fact muscled the ball in the Exxon Q 1 direction but rather the whole property and that the ball is back in all directions on the 2nd Reading Life we want to make is that following the end of it if you sold on 10 remarked the tour is that it is essential the is quoted recordings when defining the but the it an examples charmed the country 1 of the things OK so that we see them as well you would ask quit it possible to remove almost 0 if if we choose had scored in a 2 x 1 annex to the positions of 2 different particles was the answer is no and he is an example of cannot choose I just want to talk to you see that wasn't considered the Hambletonian age quit his mind as fixed 1 -minus extend that people then what is the fuel for this well it's sort of been independent index slumping 1 from explained the tools of just draw because what happens over here I will the floor locally like this they initial mother this is how it looks send particularly you see that it squeezes the it's 1 direction at the start of the new should not universal over here than this
time goes by it's going to move in this it's like the complaint and similarly in betwixt it's acceptable to China and include squeezes the excitement expert direction of so you can program was here with these coordinates for the Senate but then you can ask Well what if I use the position according to perform particle and the moment boredom and not the particles would I have squeezing events and would prominent squeezing them and the answer is is no and so cannot choose the takes 1 people in simply take the Hambletonian could be called upon to play the singing it's slump you want but the reverse time would people right now it's quiz it's 1 for exactly the same reasons before but it does this mean we exclude people during take a that with ball of initial rather than as goes mine is 1 of squeezing the during but it get talking about that's what I wanted to see world I know questions relate move on to the analysts said the itself then on innovation integration cut would consider is the following on from the Washington you expect talk most of last year or fewer In the absence of any of you with the consent of of the parties in interest in the western end of solution to the situation is a complex function of time and space time is spent nearly all the Pentagon policy this week and I want him in space for comedy damaged or this is part that and the dimension of the accident would be regrettable but was a composer known as lamentable but us this is an opportunity came along ,comma considers the case OK now and this equation is Hambletonian the Hambletonian it's age of human being the goal of operating and performance quickly and that's 1 of the commuted the last but not more and a synthetic form whispered to each analysts to be hammered poignantly descended poignant the falling all the time to find out what they have to stick to what the final cost him to find the material that has only got all of you in the right so remember what it was it was known as the majority party all the inner a product between England and the In the underlying simply because of a space which for us is that true already products was senior in the second-round win over the bar from you to and get rid of this month screenwriter and reckless president Reagan part of you the ball GX so what do I want when I want to know the lifting of squeezing result what analysts the 1st thing through we're going to be the only if we have a whole community center that is the operators on living inside you simply take it specific to another cylinder seals ,comma on Friday and artists exposed prime following the complex numbers L is an element of the Huber space which is normalized could be walking in and what that means is that the function of the penal code the property that they ,comma products between the announcement of the all-male analyzing political figures on a round of golf game we want to prove that if was looking at the fuel kind capital to the meeting to some positive and this field bank evidently maps of the ball inside the cylinder then unnecessarily a lot is be good equal then capital exactly the same statement as before now we see that even by stating the problem we immediately uncovered an issue that needs addressing plainly what what area during the we wanna say that for all initial not in this bowl we can define the
floor oftentimes the right and this was just some arbitrary bowl so the question arises when analysts polled will
oppose all know too high but nowadays we know the question the answer to that question the very well and so and so is whether .period held to concoct precisely when he she is more equal than I would been to mention do so how does that would let me just quickly universe if he is strictly less than fought over the then the problem is separate and using contraction mapping together with 2 estimates 1 constructs of solutions right locally in kind the timing political as being pounded from the body and they find what deal more of the initial OK so using this handy division immediately constructive global in times of motions in particular the flow is going to be defined weather for all kinds to so how do you do that would started initial about that he considers 0 and you run contraction mapping 3 those 2 contestants and use all the problems services conducted some kind of what was in those wishing to pay but no new sample your solution here has exactly the same lost right exactly the same or is it the original time slot as suicidal contraction of lower here and you sort of 1 another kind of what was considered among and constructing global now if problems that by working his hold over the then the problem becomes critical and then in this case that kind of local posing as Fallujah consoled the question is
a function on the notion that by itself the skating symmetry tells you that this time can also depend on the Opel normal danger about it depends on how concentrated initially that I use more concentrated initially thought that the short at the time of exist so this naive procedure which is nominally of the struggling time solutions because made with everything the irrational solution is concentrated only sort short amount of time and by the same procedure will not help from global solutions saying this case and global isn't as indeed result from Paulson and says the following think L Ford read initial . com neutral to the group but there exists a unique global solutions you want go to must be taken less so I didn't feel like slump last year include affordable consuming nations ,comma please you the the end of dissolution of these uniforms based environment so the integral part our miracle for quality or yield to the politics of the last 2 the extreme heat this bounded by concentrations only on the mosque on the motion the same particularly this theorem allows us to consider fuel was for monetary amounts of the you I guess this is only I thank you please so I To make life interesting women to consider known squeezing her results for the most critical problem in the field of flamboyantly the we must quickly for the land and the people and operations symbol for is 1 of the new that CU analysts who dimensions which is what the film but also the method that I'm going to describe applies equally well to all the other dimensions a case of well-known for us to just which we can fall In the end the theorem I want to talk about calling this joint work with :colon and shows junk and says the following things all assume that you're given a bunch of different there's so foreign capital on the wasn't number finite and the start of the season and will currently in the amount of local factories the plan won't on Friday the complex against tears it is a positive number of practical schools declined to give the fuel at times capital he pulled the ball assisted pulled in and who is mapped and it is still leads in strikeouts in and then the necessary later on in the world then kept them off no the incentive for us to consider this problem came from all attendance of talk during the fall semester innocent lost form but talking each Donna Mendelssohn presented Hudson thinking on squeezing result for the Cu Chi angled integration of the three-dimensional tossed the idea of several questions from the audience about the existence of a syndicate squeezing resulting in thin volume and at that time until this theorem all the existing to Becnel squeezing results of water on thing the periodic setting up an organization that is it was the reason for that so that's interested in this ,comma particularly wondering whether these an interesting obstruction to proving that evictions whizzing resulting in a volume was that suggested an artifact of the methods used justify OK so very quickly review with a bit of the street market in the city's 1st create something squeezing results for the 20 and Pt is due to coexist people proved to be most pleasing results for the children of the former union parts where the Union parties is assumed to have discrete spectrum and justice in courts but retaliation and he offered examples of such flows around court then the next entry in our history is going on I wanted to contribution the first one hour it was that the plane which she gave morning samples all flows of people on the books instrument and another 1 in which proved North squeezing for the and unless all the tourists complaint which does not fall on the books since many team quality and accused of lining up copper and coal proved simply convinced we can for keep the car on the recently these abandoned people of Hong and so the quality and wanderings who are the only problem this result of the I dispensing with the use of the Muira transforming the also using picking on squeezing quota system of coupled to give you the equations again on the way we have the result by remained who proves inflicting on squeezing for the Benjamin Bronner Mahoney equation of close
ability of KGB again on the Taurus by proving that this equation holds under cooks in spring water and then I'll finally emerges from here we have resolved by for the Cu Chi triangle occasion on the 3 damaged talk no don't Mendelssohn's resulted is a critical resides in the same way that ours is a critical results with winning by that that angularity and needed to define simplistic form coincides with a scaling critical regularity for liquidation the case now in that case so we only know your local will pose as for solutions are in the critical space right with the time depending on the profile of the show that comes so in order to prove present Bicknell's quizzing results for arbitrary time the eyes she assumes that the CU languid integration is globally would post complete with uniforms based bounce I she assumes more than that she actually assumes that various frequency truncated versions of the equation .period globally with ballistic uniforms and balanced that assumption is stronger than you and issue than the initial assumption would bosoms for the frequented rocketed equations in lies with and for the Cu Boston but the reverse is not true I think more about that later because it actually has a bearing on our animals were they are not what is special about this .period and the formulation of the political as well as saying Give me a time and I want to through lower ball inevitable rank them in its U 1 100 or wanted southeastern Wisconsin it is true that are willing see soon like that is not the case in the critical case because Skilling I don't I have told you didn't do it uniformly and globally in time on pointed that out when it going because you need to do stability of your final dimensional approximation in order to do that and have to work along with the time much of it so um "quotation mark station of other people more very briefly make the setting so what wanted didn't do is still the solution and expresses as a superposition of language that could just stand on armaments including existing 28 on something over carries on in Zuni you have to to clean usually the so even if 1 takes the solution and truncated tool employment anything less is that 1 of these appointed dimensional canoodling insist that translated to 2 frequencies a small loan including land either by using a shot projection in the case of a garden order a nice smooth projection in the case of the idea right there don't and gets on fundamentals "quotation mark on fine and emotional systems almost applies and so the grumbled is going to tell me that all squeezing holds for fortifying animation system so all that 1 needs to do is prove that solutions to the finite dimensional system had a good approximation to the solutions to the all full equation in order in the news than on squeezing being placed on squeezing the complaint the right to know of moving from the periodic sticking to the to the European setting it's not clear how 1 should define a finite dimensional approximation In fact that was 1 of the key challenges that they had to overcome rightfully defined by intermission approximation but moreover because the Laplacian on Euclidean space has absolutely continuous spectrum 1 cannot even find of finite of dimensional sub space of to that is left invariant even by the senior float all right so what did we do when just sketched the we are contradiction also assumed that the have parameters the theorem but we take a lot to be strictly listing capital on hand could fuel a time capital of the maps is more inside of the innocent polite to derive all contradiction so what are we going to do what we are going to use frequency truncated large books approximation toward OK so I'm going to choose parameters frequency scale is going to be in the media and spatial scales going to infinity I'm going to consider the following find dimensional system according analysts and identity that's not lost in the way of what I do and I answered the projection in the morning frequencies more than and I gently give the singled home the old my solution 2 frequency small wouldn't 1 to make sure that this is handled .period right on the Hamilton in just a 2nd OK where do I pose this problem will still going to be our but critics is commentary team which is asked to do more and in the end of the 2 companies this problem on
ever-larger tore I a gate I take more and more frequent as a project Ohio high frequencies and I pose a problem on larger and larger
port and I'm taking my initial . com you can find 0 Caribbean you 0 in on a function in a chance what is this these are the functions of a failed to all this large tourist the property that they do not have frequency larger than women the I have made I don't want anything from citizens of Hambletonian cistern marriages break down with the flu and the poignant these things and he went heart of arrangement of humans could can't come from what people told the project should be consistent with the him and he went to the powerful no because this is a finite dimensional Hambletonian system of almost squeezing theorem applies so 1 can find witnesses on squeeze so I can find you 0 OK clearing in the old friend and the stock of previous heart when his ball using such that the solution corresponding to this initial doctor told analysts in a time capital city lies outside assistance when I'm going to take my cynical to be a little bit of mattresses has become intensely less on no I use it it is a little bit of fudging going on their right Z's star an largest some elements of failed to win rights and particularly of his always the
star should leave although here however I can take this the starter and projected to frequent is more
equal than any in as little interesting EDI making the small and small error by the world is the right then at some point because I have a positive distance between the 2 laden capital on your iPod and and fate of at some point that error becomes acceptable OK so I'm not gonna write the projection of hugest just bearing in mind that there is 1 too many things perfectly To similarly overseer right by making unacceptable air I can at least held by a completely supported functions and as the infinity Woodbury eventually leave in the domain of 1 on 1 with the of the tourists case so what I wanted to I wanna take these witnesses to know squeezing and I would produce a winner small squeezing Ford always equation the CU analysts on the on a lot games so what is the strategy naive strategy is the following take initial it's a matter by analysts and put you in mind and let's take quick limits private cover you zeroing funicular he and ivy-covered you team thinking over and let us assume that we can prove some that these 2 solutions aren't really looking analysts flopped because mining by that that this function is the solution to the jury again unless 1 2 with this initial doubts about time capitalist please note if I can do that then I'm done OK and the reason why is because the complement of the cylinder and the ball out close on the week limits OK so these being the winner system of squeezing would imply that this is a weakness will squeeze focus all right so I can I make this strategy work you have to think about it for a few minutes you will see that even the simplest step of passing to the limited extract Newsweek limits over their eyes and the controversial and the reason why is because every single element in the sequence in a limited set of leaving different hubris space by the once about for 1 102 told the Taurus and the other 1 is to all 5 to so called Dwight House the week limit while the fundamental until have a function on the Taurus hockey newly invented to function on the please understand that this afternoon and this season would be neutral not using it well if you want to do you cut the Taurus your racket thing him that the solution in the plane so think I I really have to be very careful you you cut to the Taurus will you cannot cut the tourists at the point where it seems that the usual but has a bubble of concentration level of mosques and the reason why is because got it over there and you want rabbit won realignment of you get to locals still have lots and the solution to the equation with 1 bubble on the dollar's looks nothing like the solution of on the plane with 2 home balls OK so that I have a lot of trouble proving but at the weekend it is going to be a solution to analysts OK so because of that we have to go at the point where the solution doesn't have a lot of mosques in fact for the same reason that I just explain not only do you have to cut it at the point where the initial doctor doesn't have a bubble of concentration but the solution of the thank kept this team doesn't have a bubble of constant for exactly the same reason so I went to court when choosing the Taurus to be sufficiently large private thinking and to be really really large and using the pigeonhole principle I can find the regional the tourists with the initial not boss and that's where women come back 1 going to prove that was that they have won almost finest Der Spiegel propagation for the equation right because initial letter doesn't have a lot of most of which he is the solution occupying key is not alone have a lot of Mosul overseas in order to do that too limiting their good Torres to be really really large compared to the frequency scale any activity in the area and is going to be much larger than any time so that the solution doesn't have time to wrap around the poorest and most new put a lot of us over here 10 the infrastructures of the tentacles the huge of With the velocity I have of the projection remedies a projection of because more equal than an end in the morning Eric the jury adjustment LAN is going thinking much faster thank you that's the whole point Ellen is going to be much much larger than an intense the bank team we won approval squeeze the case are so old that they scaled the weekly limits contained not the next issue we we have to deal with it's the beauty of this is where I grew oxygen masks going so what do I mean by stability of the world finite dimensional approximation I mean that as you increase the quality or pure approximation you get uniform space that bounds for the solutions right and that is going to give you a fighting chance to prove that the limit is 1 of the indeed of solution to analysts complaint know this that's up to the ability of a heart problem because of criticality tends to make the discussion a little bit simpler let us consider stability this equation where we dispense with the Jorma treatment rather than you are asking for this equation on the poor writers just considered on Euclidean space and even then is known to increase let's us let us also the VAT quarter analysts and other clues :colon on 2 come to think Henry sold this problem I now because the situation has escaped symmetry Uganda you was that skating symmetry and using that stability for analysts and 1 of 2 easy equivalent to uniform space-time mountains for the following equation like the place on the N N Y 1 and of course in doing so there is a vestige of the crew of the skating that I just use and that is that the now I have to find uniform space-time balance globally in time program Duncan refined uniforms based
calls for this equation globally in time but if you look at adults in the era of it he did it with a cubicle or doing He's ending a projection to low frequencies surely you should be able to prove uniforms for this equation out that this is a strictly Honda immigration along universities demands for this equation actually implied Dalton's Spearman the converse is not true and the reason why His because 1 can invent solutions to a so that to the usual cubic adolescent play the colossal solutions to the secretion by using skin OK so that is a fine you cannot use doesn't guarantee victory hobbled approve in your rescued approved and the answer is no and the reason why is because these projections of a senior destroyed a more it's more often City form great the reason why is because they do not commute with weights the Horowitz for Milan compare there's no reason why the commentator should be small OK so I am all right the previous office also had to to face the stability problem why was it easy of use for more than 1 thing where in the subcritical this might let's say these well Posada's all of the 48 immigration actually the same as that implies with pose and suspicion bonds for the full year for the frequent attributed equation basically because 1 has air a luxury awful holder in time talking when deriving what what about the media other critical resorts right what the Mendelssohn well as she assumed that the frequency truncated equation is globally will pose we've uniforms they not spent this is 1 of assumptions so what do we do with our well we developed a method to prove uniform space-time balance OK provided 1 modifies and these projections slightly so what is the modification there are no longer they usually to projections but rather they are projections that became very very very slow pace but has the advantage well that 2 cases you can consider either from the initial doctor is very localized in frequency auditors supported on a very large frequency bands and if it is very localized in frequency and then it meets these projections of basically like coupling constants because the very very slowly and because adults and gives us a reinforced this time bonds forged the you begin this weekend than bonds for the CU analyst with the coupling constant OK know what happens in the initial that I supported on a very large frequency band well if the initial doctors supported on a large enough frequency band then by the pigeonhole principle somewhere in the frequency band I find a a region where you have very tiny mosque and I speak to the initial got back into true bobbled straight 1 to the left all the tiny Monteria and 1 to the right all the time in Marseille no we use an induction on my side demonstrate how does that go these events once I have led a spirited them idea initial got into a tool to ensure that each of them is gonna have Maastricht list and your original must started with so I can solve the problem globally in time with those 2 initial . com and I have uniforms they stand up now because the twin show that I would win separated them frequency initial time right you can prove that the interaction between the 2 global solutions are weak and whether some or all the political solutions is not the solution it is almost a solution to perturbation theory is going to give space-time loans for the solution to this situation in that case debates induction on mask when it supported on very large frequency bands and simply don't sincere and when it's very localized in the compare all right so that's what we do we come over here and modified its projections include by scheme the so what we did so far we obtain a stability for this equation but outpost October so what I have to do what I have to take this solutions that leaving even the plane and I have to wrap them around the ports OK and I have to prove that they are approximate solutions to that problem but now in order to have a perturbation theory good enough to do that venue that I have a pretty good idea to have perturbation theory in critical space which means that they need sticker system at the critical regularity helped to write critical stupidest however gun tells me that the face Nano's tickets estimates at the critical intellect OK so what do we do when we proves to jettison its and the critical regularity but they haven't been adapted told parameters and and any strike if you take an end to be much much larger than any than the solution doesn't have time to wrap around the torus Inc improves to consistency in that the industry consistent refusal critical Street a distance on the 1st all right so my mother the top of the line can have 2 more minutes to finish in the right so that the very last thing we have to do is we have to prove that these 2 limits over the island village life in the less likely they have that all we can limit of solutions to analysts is a solution of strong solution to analysts right so what we need is poses in there with quality of now think subcritical say things like poisonous in the wake of pointing construction work of powerful thing that critical thinking the only result of your able to find the different chases result of of lesion on improved logos and as for the energy critical wave equation OK in order to do that they use the concentration compactness principle they developed for that equation I'm just as the one-minute explain to you why you need something as heaviest concentration compactness to do that compared so consider the following scenario that said adding frequency you have forbade a lot looks like 2 locals once supported around 0 and 1 supported by around 20 per cent in investing in the new boss the limit by the weight limit in order ,comma because this crucial that other lives maturity by 2 leading dissident Buddhist anything compare now in the subcritical skating consenting the high frequencies are we in particular their small in all the relevant norms and is no hard to prove that the solution with this nation that Beckham would just for the solution to this initial last week From now in the critical sitting on frequencies are equally strong right it's say particularly the solution can the investigation that I can have half its normal procedure and half its normal over here OK so in order to prove that the solution at this initial that would weekend of the social definitions of what we have to prove is that at some point the solution to the problem with this initial but that is the scum off 2 solutions 1 with disillusioned and 1 with this mission not vague and this is precisely what the senior provide composition that Gates it's it gives you an asymptotic principle of superposition foreign anemia equation no 0 we also use conservation Obama's do that but you know something is the more complicated because there is also a change of retreat right this problem is of course on the forests while the problem was the sole disposed on the whole plane Salazar John and of course there is a change in immigration because the GOP lost in on the tortoises nothing little about him but in the end of the world well ,comma moms which those flesh with your heart you refer to compromise to prove wrong who's 1 of whose gurus was ruthless groups alone and sigh that was actually the original motivation the coaxing to prove his results right he was he argued that a denounced the exterior measures in a way how weakly turbulent of fuel can be righted proves that the energy analysts need that created the low and the frequencies by thinking cannot Toronto high-frequency hearing 1 of you is that we're not uniformly on balls thank you hear a lot things of things we think that this is what we continue to more it ,comma it's clear all notices In my view of the state can uniform remains so a little more than this loony was facing his firms all To do so water introduce the projections on the part of all firearms is worse than little you allow low-frequency was used to be so all of the or no hope the 3 of them have been handed over you belong to assume a review for the rest of the world what knows ruse mark from our uniformly Koreans youths and Interstate Eltopia a character and you get to that statement considered the for 40 just when all the functions at a frequency and that it that comes in the footage was 1 of the function at the frequency lies within a lesson that will threaten their within hours circle small circles I'm going to get interested in whether you know can you say something bad about the resolution of a single frequency even if you're willing to give up on all the other frequency and Yancy is no right or not uniformly in all these years so over the years the ferocity when asked quantitative more quantitative and In what they wrote but you can think of this result being on the plane and as a statement about not about scattering as well as to whether scattering into attended this aggression Scappoose adults assured us that the could begin as so intuitively it means that of the energy believes any complexity just runs away to the when hours with interim says is that it cannot do this uniformly on Wall the question of Hong Kong this is the greatest last report suggests that instead of inventions Christian at on the end of the same and using digital sales within local media said the move was a very way to the outgoing waves red and you know what you have quantities a few the sometimes interval scattering for 40 years of his career including schools so the important thing is actually not using should not just be a wall so angry him and his party as use also were general which is the last possible what oddly cheerful to define a comparison data capacity and implement 10 during a vital that doesn't make Jonathan them and trying to do that for all her Bloomberg you I can't say anything about arbitrary since In the other question for With the the few and