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Global regularity and scattering for energy critical geometric wave equations

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Global regularity and scattering for energy critical geometric wave equations
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I will present recent results, and report on some new progress, concerning global regularity and scattering for geometric wave equations in the energy critical case. The emphasis will be on the Maxwell-Klein-Gordon equation and the Yang-Mills equation in (4+1)-dimensions. This talk is based on joint work with D. Tataru.
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this aggression on and 2nd the news saying that highlight a list yes yes yes OK so I should also say that tightening of adopted on the time I should say that on this formula is actually rule only narrowly chose because on the way I would the wait on the sea on actually goes to infinity that about the boundary focus on you need some legalization procedure but also ministers with the on figure by the the message of the 2 of you hope be that on that suspected thing that work the big but on British soul what you get out of it on 1st of all is the following that's on you have effect .period independent music improve on that this is on a roll these intervals the following is funded from the local both by Hong Song supplies and find the energy focus of this something that's all that's always found all in the CEO and the on its approach to security and what this tells you is that this space time in a row at all is finite all the road and this all is this signifies Feisal look there is a on the way here which causes 0 on conveniently on his approach the great so this signifies that the the fact that this new rule is finite tells you that there's some already decayed .period on this integrin as it was 0 right otherwise on you on this cannot be so this is actually the on the team crucial ingredient for the on portable extraction as operated by now so what this tells you is that morally the during this sort of thing like you know on durative on on in so many words sense as a 1st appeared right so you would make use of that you say that no 1 can extracts they all the solution you know on to limits falsehoods fall sequences solutions and say that you know that was the stationary bike that's on due to the decay all stood at all which which comes from this right but then you have dividing into on cases 1 they prevent such also realizes that there is this enters the on as you all of this but the field is equal to the boundary of the call but so this is only effective arm in a fall in a fixed your time home not in the hole ,comma so you have to cases 1 when there is on when their sustained on fixed Tenneco's gentle on the field of footsteps all song luncheon aluminum energy remains on in the panic this should be all parties 1 and on your that on 2 of the cases went on this doesn't happen by time acquires meaning a clone with opening which is to be smaller than than than the snow in on the insiders lid from the you can also so great so I only concerning on energy on and and this on the music so that's what this is going to be with his friends say that this is the case with the song energy on remaining in a fixed on time a copy of it on and this is the case when on the exits all on fixed like panic and on the news as saying that's all so what happens is that in the case of 1 the decay all of such a on Pennebaker field of disaffected again and this allows you to go along find the desired sequence of Long Beach time scales which it gives you the whole structure you see all this explains why you have this particular on direction of stationary freight because all this is that it was so imagine that you found the sequences right on annual imagine that's all you can do it in such a way that on the same incumbent on the case looking on in these on boxes and then you realize that on that experts on along with politically because this wider Phil might be the on the direction the next time I factor in the on the direction of the school support at home and like me that and but then so game on the basis of the political use of it in the sense of right but then on youngsters right right but then the pointers on fact it so you can you can actually extract such a sequence by king who had 1 of the tools we wages stresses across if you if this holds for any fixed of anyone you know any fixed on chemical so the case that they cannot really cannot use that mountains a formula is exactly when you cannot find such a call so in other words the energy exits every time I 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of trouble it's through life in fact but you can do this you cannot look at on this day this statement that all this waited on court appointee is always it and realize that on the tests were very good weight on certain components of the the pretzel in particular to the leader of the taste endangered of the text In this tells you that in fact on there cannot be any on malleable a more positive than say the following facts all is 1 way to quantify this revised so you 1 later wall fines a concentration employment contrition would be through long look at and on your past this law integrated ensemble Over the years on London Gabaldon awaited so that on its admissions quality so you wanna take something like think on this and say that this is very small what is aware until I didn't think 800 has also I think of this and on this the music that I told you on base because to the statement that such appointees uniformly for all on London and all ex post 0 as you approach this is the time to get and of course was happening is that no outside the because of anything that's happening inside there's no is chemical and on and read into this fall and there is no Money evolves after smallness of there have been United's which did you on this and the key half come all the way from Korea interim loan that he proved precisely enters at this point so all the result is that if this is small enough so that we can continue
sojourn on past 0 so is a contradiction in this fall's so this is a contradiction in this all means the summary of the probe I think I rented since I can't think about him was also stopped he and his family and so that would lead to the loss of his formula in which itself was not among the artists had no idea that desert is the view of this the only solution is to use the cases using it going of Yunnan was because they said the yard so it's a really it's a very weak decay of your statement that it was read on tells you is that on the song this snow along this funding is founded right so obvious tells you that sort of form the the interim actually decays and some already stands at the center of the field and coalition soldiers although so what you can do it all because you know that all of this is so I am saying that finds the Burmese official but this is all responded and this means that you can take care of you can take it to the to 0 right all it's usually don't and so on they will tell you that this is terrible on spot this is the 1st of his his yeah yeah but it's only a in it's on the left side right I think that this is uniformly bounded for any future that's something that you can prove all the improved so you can take just the 2 of us yes yes yes the completion this is the 1st of these is that the Americans who all have had and so on and so and so of course but also and we also have to use fall sort of on by the nearest minute stand and so on so all by the complicated because of all the way to the same thing with now that's a very good point so in fact because his guests news Of the releases of Speaker of the National so Soviet he's more of a yet so all died rare exactly so to see on the field equation if you stand has a on a term which is 8 times during coffee and I'll you rights of this is a complete fidelity for on 17 years for the apartment linearity because in the end there which they say is that of pneumonia he became circuits to say that on you have all you have very good also diagonal decay in you decompose into onto the key suspect in the point is that this on translates to a smallness of some vessel more yet and that's it but they share it you realize that because of the presence of jury and way Britain gains you wonder if on there's no expansion decay between the low frequency partners and then the height of his powers so this is actually the difficulty the ones yeah that is positioned to actually I think the problem that we had to do was this what we have now as soon as the group that has been used approved have the government whose you say that you get the you know that the people who the right right but then read right but then that's when all you you construct a minimal all kinds of rare so right I mean I guess the point is this is more a more direct approach so we can also mean that we have a minimal kind of resuming the canvas any kind sample that so far this year more than this the use of continuation of course instead we have said we have said do from the outside outside actually been because of property of all and false details so it is very easy this you hold on to the market in the language the continue the minutes a