3/3 Obstruction Bundle Gluing

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3/3 Obstruction Bundle Gluing
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Hutchings, Michael
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Institut des Hautes Études Scientifiques (IHÉS)
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There are easy examples showing that classical transversality methods cannot always succeed for multiply covered holomorphic curves, but the situation is not hopeless. In this talk I will describe two approaches that sometimes lead to interesting results: (1) analytic perturbation theory, and (2) splitting the normal Cauchy-Riemann operator of a curve along irreducible representations of its automorphism group. Both were pioneered by Taubes in his work on the Gromov invariant and Seiberg-Witten theory in the 1990's, and I will illustrate them by sketching two proofs that the multiply covered holomorphic tori counted by the Gromov invariant are regular for generic J. If time permits, I will discuss some ideas as to how both methods can be applied more generally.
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it so if a top and and you young and very thank you so I guess this is my last chance to try to get my point across so before getting buried in calculations again when we try to explain what the general idea of obstruction this is supposed to want little bunch of things together usually will talk about holding off a curse although it might be examples of our more Mauceri and this was I have a bunch of people you either 1 including together but now so there's some marginalized space and so is my joy space and consists of but configurations of the new I together with some going predators so general these home occurs might themselves live in some on spaces so pretty for any collection of Homer occurs in their respective moduli spaces and then some gluing parameters singly come translate things etc we can try now the best situation where the all of these linear rightist operators city isolating arise equation for you like the best situation is when these operators or surjective then there's no obstruction to glowing and you can just use the usual construction everything's fine the 2nd best situation is where these operators are maybe not surjective but the Co colonels have constant intervention is as you move around much less space dimension of the CO colonel was sort of this operated doesn't job so that I have an actual vector bundle over the space and configurations with the fiber over .period inspector bundle is the sum of the CO kernels of all the operators for all the pieces this is when we have a well-defined instruction bundle the worst cases where these Co kernels Co Colonel dimensions may job then nearing current EC way under powerful there incident so that's when you really need to do something fancy so this destruction bundle case sort of in between the classical case everything works fine then the general case really need not really high technology needed to solve the problem OK so now so let's consider this obstruction Mundel cases so that you can try to gloat in right down the glowing equations and there's an obstruction so you can glue up to elements of the CO kernels of all these operators and so you can then define obstruction section s so as inputs cooling configuration going parameters and Alpwoods the obstruction to glowing whenever equals 0 can glow and generally can prove that this construction gives you buy ejection from this year's sentence as to the set of possible and then you want I understand what are the 0 suggesting this section as defined some strange indirect way finagling equations so you want to approximate as was something easier to understand so S 0 is this linear iced section at of the act of an act which is obtained basically from taking the lead in water terms in acid throwing everything else away then you argue so if you try to do some counts of the number of 0 suggests there a general this model a space and is going to have some boundary so so the glowing parameters can only be in some certain range on the new argue that on the boundary of his modulus face the terms it s 0 are much bigger than all the other terms and so if you linearly to form as 0 0 is never never crossed the boundary so many countered any kind of counter 0 C can make for the same for arsonists 0 so the neatest have to understand this year's 0 and then the idea as 0 I think they didn't explain this so well last time so many try again on so for each of these small a curse or whatever they're trying to glow there's a component of hero in the coal kernel of an operator and that the contribution so so they component that is 0 4 1 these 1st has contributions for all of things that are adjacent to it for all of the things that are being glued to it so if I look at this you 5 what is the U 5 components of this section so for each of these curse line you to the new 7 there's a contribution which comes from the asymptotic S and tonics at the end of the current can looting you fought so there's a contribution here from the asymptotic so this and if you won the contribution from the asymptotic so this and if you too In a contribution from the asymptotic 777 on to basically look at the asymptotic expansion of his curve you can write a kind of free expansion of it in terms of IT in values of the linear operator you look at the leading order traumas that she's realized I advise the asymptotic operator you look at the leading order terms this guy hasn't exponential decay so this contribution is gonna look like you the miners are under Berlin the Eigen value and are asking about you translate this for the that's that that's the basic idea and then if all goes well you can then actually count this year's address here the figure out how many going somewhere any questions about this general picture before plunging into minutes example yet so what do you do this for a new set of coordinates and that we political direction using this corresponding to that of the highest in the US and then do you want but I I'm also called the win the union said In this if you look at the college in the fall ,comma from 1 part of the use of the fulfillment as 1 of the local media that the connection is to hear that the people in the community connecting to the top of that Goldstein things should come together police are really interested in is a sort of far away from each other the loathsome due to the images Sylvia and fly giving both to win doing the whole building at once so we're sort of pre glowing so he is in this example of God I've got 3 levels and do these so translate these guys up but some large amount they can move but different amounts up to translate this 1 down this amount actually these guys can also translate up and down with respect to each other on they preclude using some kind of functions of the native I deform using sections of the tangent bundle of each these curves and then I saw all this whole system of equations for all these status be 0 0 or at least of the comparable the operator someone that will all go to the interior you but nothing seems to bother the "quotation mark although there a little bit change from my women section side 5 which describes how much a moving you 5 2 in permanently for that reason all the pleasures of like couple of this guy will kind about on his account of 1 should be fire for you file which is supported on you find dies the dies answer the ends of the floor and as a function side sci-fi which tells you how much new moving you file so so I am moving the whole thing and in the end of the day of the 1st day I understand that there is not a place for sci-fi conflict which contributed to this and all will see
that what has already told the what your only and that was the other way this site more blood than I should have listened to completion of the deal this August this year some of the numbers for the others Members Only this year but I don't really quite understand it or if you're happy unhappy but and other questions in the aftermath of the equations of last 7 the what's so funny so whose office includes a look at the whole if I were you look at this more remote corners of the truth the ship the resolution of the object is to realize that there is no more you should be well it's not fundamentally trying to do a mean you could shift the different components of the subject of the loss OK OK so so far from the world was more of the universe me whole book market value of the the Rule of Law Review the problem was that the familiar will about this it's all you would do something was refused to use all the cool because it sort of depends on what problem what exactly would probably try to solve the you want good things know well what what exactly what kind of equivalent masculinity I don't think there to both Helms point of view yes from my point of view there'd be a more because the B 1 2 3 4 5 6 grueling parameters but I'm sorry more attendants level his followers say but I don't think so it's a little more than the rest of the world that was the last of the open we are at the time so this is not find useful from that point if you can't solving trying to solve that problem than there just to impress millions of Americans with so many defendants the get your name directions for the world in on them and then will have to face you also need an to image the area also won a true that means and each of these pieces was in some bodily speeds which is not operating so this this you finally listened to marginalize space of unbroken things words of his earlier this like it so this whole configuration is part of the boundaries of so much of this will try to understand how many ways can we glue boundary corner configuration is something in the interior of the modulus all right so let's let's do something to talk about but the 90 generous contract 3 default sponsors have been the venture field are on the generic most complex structure but our crosswise satisfying the usual conditions which I wrote down this time Ronald we wanted to hold off person or crosswise on so talk a little bit about pressed covers a trivial soldiers To suppose it is an elliptic report simple recorded hang here so it elliptic mean well all years my work gamma it's a periodic orbit of the vector feel are then you can look at that on normal bundle to this reporter is actually true identified with the contact contact Plainfield restricted to this really working on In the derivative this flow defines a map From this normal plane at a given point and gamma 2 Celsius look at the derivative flow With this the defines linear wraps Mussolini rise recurring From the contact claimed that some say 0 with has a simpler to form to modify the decree under the linear return a two-way simplistic matrix 90 generate means it does not have 1 in its spectrum an elliptic means but this mountainous continued to rotation while some angles to theater the fate of sort of like a rotation number the sorbet and not a generous C of the war-battered and all this multiple covers tells us that there is a rational and we had the trivial cylinder Clark crossed In our cross-license licenses of all Norfolk cylinder because of the usual conditions on Jay we we want I wanna talk about breast covers the history of stronger but this lost the previous we were just talking about the 2 of you this it's a good but by the time I figure out really where at the time masterly stroke costs will be over barring so I suppose I have the branch cover well agreed to Grant cover the cylinder Rossignol was saying has testiness G was connected into history I must say it has a positive ends of multiplicity A-1 through it OK so words if I want to use our Canada but I have this service that's that's covering it on so each end of the surface is going to give me a covering of sort of end of the cylinders multiplicity so positive ones but multiplicity is a lie the negative ones the multiplicity of the winding through the L than the sum of the eyes in the sum of the PGA's must equal the because of the whole thing is too greedy so
this has public structural pulled back from the complex structure of the cylinder This is a perfectly good hold off a curve it's nice example of failure of tranches salads like that To Chris Wendell did that sort of a closed version the version of this foreclosed person his toxins as but for Varela version of that on so what is the Trantor Sally Field what's again it was the freckled index of skirt so the index cosmetically 0 well in general for the index of any curfew enough or across the continent 3 manifold is minus the Oilers characteristic of the domain but plus 20 some relative 1st turned class so solicitation is so too towers a trivialization of the contact plane over all the really warrants conspiracy towers the role of first-year class the concert playing field on restricted to this review that is the sum of commentator returns she some of the positive ends the comments Sanger index of the real worry about his back to the trivialization minus the sum over but the negative ones of the comments Santander to enact a negative answer the same same formula works in any number of dimensions except the miners tying replaced by an minus 3 times kind words effects on known this particular case "quotation mark I can just use the same trivialization for every everything coming from some given trivialization of gamma and then but the decision is excited about on so tell there's trivialization of Over the report the index of you 0 minus characteristic of signal on his 1st term class goes away if you use the same trivialization of who've given so they have plus plus the commentator on so what that comes out of the so goods to G minus 2 plus capos L so that's minus 0 characteristic of the curve and then so if I fixing trivialization this then the beside the grave of linear rise return after this really worried Gamez rotation led by 2 5 faded and the college senior index it is so if I have the Sanger index again itself it's 2 times the flora faded plus 1 on and from looking at an AI fold most will cover the net has the linearity for that as rotation right in the
eye and say sorry at 2 times the floor of a I think costs 1 says the some of the comments saying the index for the positive answers and for the negative answer miner's son from trickles want at all the 2 times the flora PGA fatal plus 1 planned for convenient unafraid this whole differently orators has 2 times the ceiling the minus 1 the statement because the trivialization of I use the same trivialization coming from a given trivialization of gamma than that trivialization extends over the entire 1st and so the 1st classes 0 so the 1st a role of restaurant versus reviewed struction to setting abounded trivialization over the whole herd in the world yeah yeah that's OK it seems that the right 1 the year I mean that that everything's just mapping to this 1 reworded and trivialize the contact the Pinson although I did at the start that's not the way I want to do it was make the goods keep this the floor I wanted actually want to change this to the ceiling is now I know I can cancel some stuff out taken I can cancel this posture With this 1 this 1 making canceled plus With this place was over here sir this whole thing comes out to be 2 times G minus 1 plus some on ceiling of a I think minus some overtures 12 flora of DJ's said on and then there is a confused which way it goes but that the sum of the ceilings is a greater than or equal to the ceiling of the sun so this thing is greater than or equal to the ceiling did say Our and this thing it is a lesson equal to the floor of so this whole thing is greater than or equal to 2 times G minus 1 "quotation mark plus ceiling of 2 theater minus Florida's there and because the state is not an integer the ceiling minus the floor equal to 1 so this is equal to for to change and so was the Talisman tells us that the index but this multiple cover is greater than or equal to 0 that's an ice pack to it's not true hired mentioned unfortunately but 3 mentions this French cover trivial cylinder always has indiscriminate 0 and it can be equal to 0 so the index to the index you 0 the Daedalus 0 if and only if the genus is equal to 0 and so on this summer overall the ceiling of a I think is actually equal to sealing the fate of the head and the son of a chain of flora of future it is equal to the Florida theater services on so some
cases the index's 0 so all ,comma notorious sister ,comma for comic relief here is that this analysis senior reason jokes in between all the analysis of great so that's so why did I tell you this so what's the situation we can have this we we have a friend next 1 curfew was closing you Plus which has a bunch of negative events at covers Kim so this year plus could have negative ends a gamut of multiplicity is 1 through its head that I could have some curve minus but also from ex-wife which is positive and it covers a camera of multiplicity is PGA and then in between them and I could I I have this curfew 0 would sometimes says index 0 I suppose for a situation where this thing actually has index 0 so that in principle maybe I conclude this configuration but there was no many gluing there are so the reason why I looked at this problem the of the cliff tops and I've worked on this problem in paper was was to prove that 0 1 the differential and embedded contact knowledge satisfies 2 critical 0 so for that application you actually need to do stuff like this although the paper we actually did much to something much more general easily we would restrict moral anything of this kind to you in but for general generally than that stated for about contact mulch we also did orientations very carefully you're wondering about orientations very carefully Brazil no question the the on both "quotation mark you could certainly do this kind of thing more generally for grant covers of something which is not a trivial so we didn't do that all that's that's of interest when years China's political borders maps and I can't entomologist Chris carriers the 2 of them that's true yes about 100 paper looks at covers of cylinders nontrivial soldiers did not pronounced with the nontrivial which come up in shame talkies foreign variances cylindrical contactable it's very nice a very nice example obstruction bottles of data McDonald's settled into the we just had dimension but the kinds of configurations of included more general than what new procedure want that to many years of this the rest of the news of the so well you have to prove that the document was it that's cuts coming in 2nd quite so this situation a situation like this we have an obstruction but also say 1 with this thing so what is this what's basically moduli space branch covers so there's no denying the reason why this is an example of failure Fanta salary is because 3rd dimension so this the moduli space of all such courtesies 0 it is the manifold of ranch covers the soldier and the dimension is 2 times the number of branch parts the number of brands points is Kepler's elements to the table it's whatever this thing is not a cylinder Anderson branch .period so it's a modular space a positive dimension manifestly but it's index might be 0 sudden changes Valadez doesn't work missed is absolutely no way we can fix that they choosing generic check but some of it from so that no then there's probably cancer affects the glowing parameters bonds we could fix how much wouldn't translate the stuff out there we have a modular space ranch covers on the Net Over at each point in this module a space we have that the Coker of the operator of the 0 4 the curfews here so soon that occurs you plus a new minus kind transversely
so there so those so soon and that of the operators D plus the minuses rejected which you can do it at least to these things were some injected you can assume that the cost of however this D 0 definitely has a cockerel farm and as Chris pointed out well the dimension of the Coker never various so dilemma and while have always come which works here the story 1 more you run but this is certainly doing this problem this is not the right way to do this ,comma again generally cannot disconnected French so on having the support of the shows like centers an instruction manual for use in making the restroom is like this big instruction sheet and how to wash your hands I recently got for the porous similarly 25 of pretzel lemon is that the kernel of the 0 was always equal to 0 just as it is this is the focus lentils Torquay talked about on branch covers 1 of closed surfaces and proving that those that Colonel equals 0 you're talking about press covers of stew of struggle Saunders society difference I think so this is just a simple calculation through you it's automatic transfer salad thanks to the rear right effective forgot to tell you what is 0 it's so the deserves not we think it is the very sorry and then getting scattered here this is the 1st of all this is the sort of I want a normal defamation operator so this so this operator he was 0 it was a good step we can take sort of L 2 1 sections of all use Europol if the North bundle 2 controversial song so this is these are basically sections describing how the move might prove you 0 in the directions normal to the trivial cylinder it you think of this as deformations that don't move the pressure points roughly speaking this is the same as yeah the to Bolton sections of "quotation mark 0 1 forints on the same with courage values at this point treasury deformations adorned with a branch points so so the index went to this thing the dimension of 2 times the number of press .period but in the index of this operator T 0 and it to me so that can her 0 itself is index 0 so that the server usual operator for the curve would have index 0 because I'm too restricting the domain this operator by not allowing the branch points to move that decreases the index by the number of constraints that provide which is 2 times the number of pressure points sets within service operators minus to be on then lambasted the colonel was 0 so that this Cockrell has dimension equaled to again so I have the right to be bundled over to the dimensional basis and I can expect to cancer-fighting counters 0 which the number was too little was a proof of this some well so you let up with society and answer the colonel suppose such they exist the 1 account how many 0 it has the survive this sort of :colon similarly principle this have isolated zeros which looked like a zero-sum mobile market function the algebraic counters heroes this prisoner equals
0 With the qualities and only if it's not finishing firearms now there will be none vanishing sort of on the ends and so the standard has a kind of winding number around 0 so as you go around when the circles the ends you can count the winding number of section around 0 but there and then that the numbers zeros in the interiors can be the difference between a winding numbers on the ads so it's going to be the sum overall I have rifles and took care of the winding number with respect to this trivialization the society around the neither positive actions -minus
Central genitals wanted to hold 1 of affected him on on the side of the Chinese mainland events and then you can put bounds on his whining numbers so basically that the asymptotic behavior size described in terms of the asymptotic I invited asymptotic operator don't time explained that in detail they find that the winding number this last year equal to on the floor of States while 1 of these winding numbers is greater than or equal to the ceiling Of the so that this whole thing is equal to you're right thank you so this whole thing is less a equal to the Florida 50 theater minus the ceiling of the theater which is minus 1 cent show that zeros was equal minus 1 where are the kids have been well Cuevas reading this is unhappy this nonsense that farm where services to habitable for instruction on the way to do this to the no kernels always 0 actually the business would you go we need to see that the operator of the result well talk about the latency for generalized but we have much time is left where he served as reminders to live on the Internet and the 20 billion well 17 17 minutes from the end the well OK time flies were against money when Hurricane Fran on Wednesday Francis OK so we have an instruction bundle into this instruction bundles the high so we're clear happy over here so that we can do obstruction undergoing so what's the picture "quotation mark makes us happy that this method is applicable the unlucky you you didn't have to do this with the people greeted the papers with tossed with a mixture of horror and shopping for OK but but himself but so what we do when we do the simplest nontrivial example the simplest non-trivial simple so this is where cables 1 than anyone close to an elderly couple to 21 people speak to the 1 so curfews 0 this looks like the disappear
pants the top has agreed the part of history 1 this is happening to our rescue but it was assumed will always assume that faded now is between 0 and have this is this is what you need for the index but to shifting abide integer this is what you need for the index to be 0 if there is between half and 1 the index will be too the ones this how bred for incident is going to be 1 branch .period it could be somewhere here mapping is somewhere over there so the moduli space branch covers we can identify us with the cylinder on this agreement just to keep track of where this pressure .period as it's actually the model is based is actually double cover of the cylinder because you need to keep track of asymptotic workers "quotation mark wasn't going to them on their when you actually gluing you're not looking at all branch covers because I've only shifted you plus the miners down with some fine amount so there's only sort of finite range in which the sprint we can be so this really is actually ahead there's actually something substantive miners are part of that process was no we try to do these things so there's going to be a destruction section and so on and so In the light of day because 5 should you put us up with some fine amount a few might down some fine amount and there's always find an interval which but the grant cover so what the branch waking up the way up there as far away from the so you're writing this is really kind of annoying too sort this out and figure out where exactly the branch .period and so on on Friday is another another simplifying things you can do which is you can arrange that the bill when the arise trade flow him this is on Jay Leno while there was a finches UGA in such a way that 3 flow from 1 . Gamache to another 20 is actually a complex linear on when you do that this this to and causes the normal operation top-rated G 0 and to be actually conflicts and the operator just until linear operator simplifies the calculations will so the so the cockerel what's the two-dimensional real vectors face a one-dimensional complex vector this so I have a complex line bundle of citizens it's a natural process 1 I have accomplished line and all over this is more interesting because now so my Mauceri example the was trivial because you have the same Cochran Colonel over every point but now these these scourges 0 variants of a different color kernels so it's more more interesting bundle but I have some instruction section asks that I've no idea what it is but I have an approximate section of 0 spending time so I fixed the ruling parameters Was it was just trying to be plus up a lot you might stumble Texas so Soderling parameters like some some constant times this the following so they haven't as 0 so that's 0 this contributions from the the negative events but you plus and the 2 positive love you might on many can actually is sort of figure out what these are so superior the whole story would whether there was a result of world exceeded that as 0 well should we give these these things mean so this was called his Parsons unions so all of this together I have new CEO 1 of the people like this I think it may be yeah 1st taking undergraduate quantum mechanics they would write equations on the board like that look like that's the please because pp In the letter P would be in 3 different things said to the 3 different things all in equation all represented by the letter but I serve cut it was very difficult to deal with it within an exam when I had to really answers under the sort of played the same letter everywhere the United States of with Woods was put it will put a hat on the 7th that work the of the interview more than dynamically scoped variables some of the right anyway just as your hearts as your hat but as the what security so this could be a contribution from the positive and which is going to be some constants Alpha times will be at this rate you need to each to the Land Acts plus the Rolandas some when deciding value some operators so physically there's this year the land as means that when the branch .period is higher up sort of closer to you plus you see more of a contribution of and
inside steamy from Reese reflects the fact that as the brand .period was around bodies you have to you have to compare this Cocolamus for different elements much less space but but on the digital do babies who need to understand the winding numbers of his coconut moments as you go around the and so this the fact that that this is this term is here represents the fact the coca farmers winding number 1 it's as good as you go around the end of and there's there's going to be some from the negative answers can be some contribution what goes on window class but for me to that -minus landed wireless the best harmonies and is what these are negative and the cockerel elements of winding number 0 says actually no teacher appeared so I don't really have time to fully explain this in full detail we get an expression like this thank you so this is this is actually quite simple but that the general case we have lots of branch points on it becomes very challenging to calculate these numbers heroes of the section renewed Olczyk contracts this case is simple enough Tennessee directly what's going on so what's going on and so on give is large then this term is much bigger than this term so in the complex plane when is as large as he goes around us 1 sections going like this the label those little here the is small that this term is very small and the industry has been but this is that this is our best he goes around the circle to cost so so that's a small and some little circle like this and then on that five-cylinder year interpolating between his 2 circles think 1 of the circles as winding number 1 around 0 in the other 1 is winding number 0 some members users at once so there's 1 way do it up to science the general case swing we calculated the number of ways to do something like this while I found had a remarkably that the number of wastes glue is equal to 1 if and only if you have exactly the kind of data that comes up in better contract tumultuous otherwise it's a number of them there's no reason to think there so in my remaining 3 minutes only just close with the question which we can think about next week the next week follows Telesur definition aggressive tea by so the city is supposed to count then that's 1 thing when 1 cursed but any through index 1 building so you could have said that the next 1 building which looks like this so the sun the some nice saved me make embedded you like embedded index equals 1 fits perfectly the this is the kind of thing we wanted him and then maybe attached to it or assigned branch ,comma trivial cylinder on say can be attached above it success encouraged .period the below it they could also it's sort of worst cases this thing could have some additional and so on I covers the same or better there could be some further Branch covers attributes owners on the side 2 handsome building were 1 piece is an embedded in next 1 for another bunch of other pieces in the but it below the undecided Witcher index 0 branch covers of trivial Sanders when something like this should make some contribution to the acidity differential the question is How much so my question is what we want is a contribution To tenacity differential so have some conjectures about this but it's not really well-defined questions were used in the definition of the Institute of the people in the market sold euros for almost all of them the consistent with those of the government and the Federal Reserve and the on what the story of the little relation to the dimensions the answer yeah that's exactly what I expect so expect this dependence and choices just to draw the picture for the simple example that I did we did this example where you're at global pair hands in between 2 Annex 1 countries like us this is intertwined with the index 0 assists and that's what we found that the number of ways to glue this equals 1 no this of configuring is building consisting of the upper piece plus the middle piece make some contribution to the answer to differential I don't what what's called City plaster and this lower piece is obtained by including the Lower Peach the Middle makes other contribution was called the Midas and this Saudi forces us model of in generally needs and ,comma trial factors in its 2 differential but ignoring that the fact that is 1 way to glue basically forces that C plus plus C-minus sequence In order to get this critical 0 but then I would expect that you could sort of choose 1 of these New what everyone and the other 1 is forced by this equation so as a more general conjectures along these lines like because the number of killings as 1 so I thought I had this is an end of the index to Marshall converging to this thing so it has to be somehow countered by D squared so either this thing they can be candidacy plus Kansas thing which is contributing 1 of the Obama signs here on or is this thing C minus times 1 so the total contribution to the differential squared a C-plus for Simitis 1 I'm really nice if I could say that we could make the choices such that whenever there's some drunk on the topic counts 0 for example or the military so what to do with the stuff from the side and only understand I was actually thinking about this question 10 years ago the think given completely incomprehensible talk about this 10 years ago but it may be that maybe now we have them language in place which actually the answer the the people of the world and the is way that the only thing that non-US weaving doesn't have to be 1 of the things that really is huge all that of course an interesting case and that we you have the right was a different questions is not a question of how many rulings was a question of how anyone account so
cities says the some numbers many defendants and choices they have the properties to give you very and silly people like me like what are these numbers exactly these examples it's not something more complicated thing when say something that I wanted to do a lot of problems including 1 that led to the deaths of things 1 of the In the event of politicians and this is 1 way to do that you have I think in that case the number of rulings was also 1 but is in general to kind of complicated commentary formula which that don't remember off the top of my head it's it's in the beginning of paper what part 1 of the 2 prepared responses some complicated ,comma formula for the number of killings right when he's let things the structural flaws that small something that there is a lot of love lost to compensate for what the American a lot of and From radius of his petition conditions which said that yes in this summer picture if you actually looking at if you plus a new minus recounted by ch then the the numbers a Island beaches are determined by deed and the and then the angle so if the angle fate is positive and very small at the top you as you said you have just 1 and in the bottom we have do you want my just love that the just 1 thing so I don't really I don't really think of splitting the middle to levels that are really look at it that way the fastest in in the world this is a Chris Carey quelled answer all of Europe technical must any also if you don't have to to questions he has a couple of sort of other examples of this phenomenon compared to tell you that so I think that's have the bottom line attention the the next speaker wants to have plenty of time to really coffee at 1 reconvene 2 as


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