2/3 Obstruction Bundle Gluing

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2/3 Obstruction Bundle Gluing
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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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but it can the end the 2 you can and and was like when I was in the free world of some country wanted to get a Diet Coke In misuse refrigerators Diet Cokes rate there but not allowed to touch cited as 1 person who also spelled out to touch it but that person can would call on the supervisors something because actually allowed to get out and bring it it's a very complicated process of her anyway OK so stamina finish with the worst theory and then try to get onto some things which heard it all hold occurs in related to the Polly fall foundations address of the which will be hearing about next week so the Mauceri example I had was you have F-16 agency is 1 parameter family of pairs of the function but a metric ton of manifold come at a time to equals 0 this trails to be mail so at times equals 0 is index 0 flow line France a critical point tutor critical point losses both in and outside call this thing is 0 but will assume that it's a sort of generic situation when I say the word generic that means satisfying various conditions which holds for dated accountable intersection of Oakland and sets and set of conditions may increased as they go along so so here in the UK so winterized operator for this use euros not subjective but when the generic cases it's it's Coke bonkers one-dimensional and the other generic thing is you can look at the so derivatives of the great inflow corporation with respect to tea you 0 I'm so afraid look at certain DDT but the greater folkways monastery 1 0 this
derivative it's equals 0 Javon non-zero projection onto this Coker but this is sort this is the the generic way in which an index 0 thinking appear and 1 from her family in Denver for tea None 0 with small this will actually be more smell do you I'm glad to the could be real value circle value doesn't really make a difference for this discussion Simmons so if you were covered all 3 of valuable Israel that what's assumption that the Yeltsin is our assumptions so Everything whole assumptions and then we want analyze is what is the change in the chain complex as we go from negative to positive too so in particular if I have to is 1 of them yeah suggest just this 1 failure of nerve Moss maleness arises at times 0 which is this 1 index 0 following appears but analysts say have another full-line Cleveland exile was not an excuse me another critical point given examples 1 the following year clause From PDQ there we have a configuration like this at times 0 I don't think it's a Welford so we expect that this can be glued to an actual flowing from Peter R. for a time small may be positive may be negative say 1 analyze when that could happen so the last time he wrote down the equations for this but so we we have gluing parameters all are which is very large and see which where the absolute value useful to his non-zero sets of values fall but and what we do is we that translates these full lines Sociedad my natural parameterize you Plus is a map for access the new 0 a map from our backs to the 1st thing we do is we translate these so we we move you plus up by Ava too we move you 0 down by over to and to the great confusion of everyone last time I continued to denote you plus a new 0 by the same letters so from translating them noting by the same letters OK so it so we translate New clause of use 0 down and that it was the year of the on the image of 0 will be far from here and so on so this is breaking a means that the inverse of gluing is breaking the what's breaking so he'd have a full line like this for some 9 0 time it was close to breaking what it does Lachemann everything's going up so it goes because up from it's near Cunard's assist for a really long time and then it goes up to pee so it's like that the sort of like a camels to hops whether some action happening and is to Hansa getting ripped apart so we took breaking through to him and that was the last and this tofu Carol no animals were harmed in the store focus or it so we translate by this autumn and then we're going to change the time to teams I'm so new it's the equation they excitability here said the team means that the upward gradients vector field for a 20 and then we have site plus and size 0 so these are on the provisions of the you plus a new minus aside clauses of section the new plus storage TX it's 0 0 is a section new
0 start but then we had a we looked at the data plus times New clause close I lost all employers it 0 please use your request size 0 beta plus invaders 0 hour ,comma functions this derivative suit of order 1 overall are in this this expression really need to start with you plus a new apply the exponential map to cyclists on and then I choose a choose caught Machado neighborhood of cues such that were both these kind of functions renowned 0 it makes sense to have the likeness of and then the glowing equation so we could call the whole thing was scoreless on Capitol this is so this is some natural artifacts and then we have the equation for this to be a full line so we have to hit us plus meets the the view as the forum to the applause science the plus "quotation mark "quotation mark minus the b 0 times there is 0 aside so as not to minus 0 plus size 0 to wear they're going to replace gets it right I'm so then were were theater plus is the 1 nearest operator you plus applied to sigh plus plus the debate is 0 S times 0 plus-size 0 plus some other stuff which doesn't really matter right at all and then 2 0 well I guess I can write this year stock in matter but we have serve plus the derivative of the ball plus the times the courage of the was expected to because the the equations changing because active fuel is changing when fastidious minus the special minus sign here and then there's some sort of their returns and data 0 is due 0 so I 0 was debated plus year's
people inside plus or minus the Times partial free to university wasn't there meritorious and then there were some letters to the lender so it's 6 see where are sufficiently large and the absolute value to sufficiently small arms and then there exists a unique assigned miners receptor sites plus size 0 such that so sigh plus this L 2 and this is this is Veronique and you have to work in some while a space where these things are required to decay on the Net is unique says that side sorry plus ' L 2 orthogonal to the kernel of class 0 results to orthogonal to the kernel of 2 0 1 the plus is equal to 0 and the the 0 I can't quite get it to the 0 but I can get it to be but held 2 orthogonal to the image of the operator to 0 so if I can actually get the 0 equal to 0 that I would have a glowing because the that whole expression of their vehicles 0 they can't quite do it there is a failure which is the state is 0 so would you think if a 0 is living in the Coker 0 which is a one-dimensional vector space so I have an obstruction to gluing which lives in a one-dimensional vector space when that obstruction is 0 this construction worker I conclude I'm in some people asked me last time why do we want data plus data 0 both equals 0 we just need this linear combination of equals 0 and the answer is because you then you get this uniqueness of sigh plus size 0 but as hell pointed out to me after the talk this is quite analogous to the on-site gluing and powerful wanna make the gluing unique you're making an isomorphism by including those keeping track of this entitling so so by making both of these equals 0 it's like analysis saying that I am making the Antigo includes 0 OK so now we have an obstruction bundle picture This is the general picture so in a particular case we have a bundle over set of all parents are Orenstein it's a trivial bundled it's it's just that the fiber or point is the cultural that operated 2 0 citizens this is some one-dimensional vector space it we have a section of this bundle so of artsy is defined to be data 0 the faded 0 provided by the slimmest so we try to glue maybe doesn't quite work so as I learned of the way mathematicians can be successful it is when you the return failure into success by calling obstruction writing a paper about obstruction but so too has the instruction then this construction defines a map From the 0 set of fast to settle glowing the self-reliance the close to breaking following a nontrivial factor which I won't attempt to explain the theory that this is only morphism so what this all could all gluons can be obtained by this construction in this does actions so then well that's nice but then we have to understand what is this section In the trickery that this were going to approximated by different sections so it what was trying to understand what's going on here so so it what's look at this this expression for their 0 no so fast Best art seems to be little more precisely is the projection cultivators 0 onto the court kernel of the 0 right so this 1st term who's in the image of D 0 does doesn't doesn't come up at all and that this term which measures the derivative of the equation that's going to stay there on this site plus it turns out the site plus is very small too much smaller and everything else and the reason why Satan drawn picture the kind functions for the color functions look like this here as in past years virtue His miners are too so beta 0 this sort of going like that innovative is going like this we can make it so
that the the region where it's actually getting cut off at the disco students they are over 6 far over 3 but this is not a Forest Service -minus overthrew the Fed and then if you look at this equation the other equation for the A-plus this thing is actually equals 0 and so on the upshot is that but away from the away from this region where where the derivative of its hero is equal is non-zero plus satisfies an equation which forces it to exponentially decaying as you go In this way so suicide plus something over here but that it it has it has time over 3 years and a large amount of time exponentially became for conservative there so the site was that you see here is heavily exponentially decaying so it's sort of doesn't matter much for this this what was secure that on this most if you will secure the 2 arms and 2 of the Flood is great so there we have an approximate section and then this you plus we can also simplify be while because we have to think about the asymptotic city plus so before translating of fatigue for TV surgery negative new plus serenity best what viewpoints of S well it's approximately the 2 the miners in which was a cameo this is this has nothing in that honest talking about more serious all that this is supposed to be a fracture "quotation mark then as she will it do this is going to give me the rescuer told me few years now threat the sorry and on start of letters you'd think with agreed financial crisis that could sell some letters to but it seems to like they already did that and all the other letters look the same as Roman letters stored a lot of after reports and letters from somewhere else my focus in this so so this is going to look like minus but this is I have my reasons for writing like this below so this word so generically Soward adding conditions to the generically so so we're under minus it is the smallest positive I mean value of the Hessian at the point .period economize here is associated the negative the floor and found Macedonian absent positive and that an 8 some function and I would assume also it is yet to assumptions that Was it was assumed that the idea values of the Hessian Q which is you this is killed His that's what that's what it looks like on and then after translating what what is now we have to we have to pair this with a CO but always translated by told translation distances so the upshot is that this is the the report times plus 1 8 1 I projectors to the CO Colonel this is approximately the to the minors window guidance all are time some numbers I can think of it as well as a number of at some that were just as some fixed element in the cockerel so a case like this did you know that I going to do this the 1 corresponding to the I shouldn't I function because this society venture this reduces streaking vector spaces but I'm sorry I said instead I in that corresponding to the site in value so I said I can function because really thinking about Holdorf occurrences system modeled his role 1st or at a minimum to make things the equations simpler on let's on I know the derivative the equation you 0 nice amorphous into the CO Colonel analysts use this map could so identified a co-creator with or so this a is
actually just a real number In a generically is going to be answer and then so I can define this what's in our the paper was taught us for the linear I section and many rise just not really very apt word because there's nothing to do with linear or nonlinear systems to leading order approximation to the section of port as 0 so that's 0 it it's now leads to the minus 11 to minus star 5 times a year -minus that's section on and then the next actually what with this windsurfer perfects the the number 0 of the actual section countered with this easy maturity was science is equal to the number 0 as part of this clear-eyed section another fact which I is of the view do carefully the billions technical things you need to checks and other now the enough factory in check is that as is smooth got in generically transmitters to the 0 section so if I want to the upshot is that everyone accounts the number of this rulings have to count the number of zeros of this thing this is now quite simple sisters the next Financial Times a constant minus 2 Loughery complicated so we can put together the conclusions from this so the conclusion this set of Soviet aid and the opposite science I the new big trouble not because there's no way this can never be 0 so that he can quote but if innovative then ceded the same size then there's a unique solution subsequent to the equation press 0 of Ardsley equals was 0 so the number of gluing is is equal to 1 of Hollywood's a little more work you could show that actually that is actually just 1 willing the was no consolation right so this is sort of what we expected To get we expected that you would be able to blue this configuration positive firing of both naturally found in the analysis actually tells us whether to you will be positive or negative to figure it out you have to look at the arson tonics of the new plus there you have to pare that was the hour well with the derivative of the equation so you can work said cells completely Switzerland forward from Wolfowitz signage she'll get it going any questions about this so that to make it harder Wholesale and that but basically you need to show that the things that I casually crossed off Our a small so basically there's this summer we serve restricting attention to some set of warranty and you wanna show that as uniform as to S 0 no 0 so the section can escape outside the boundary of this region you do that you need to show the boundary of the region the bill S 0 is very big and the stuff the Thruway very small so someone who was in the middle of the month about to get to the the end of the year assuming that the title of the year you don't get along at the top of the value of their investment worth 1 . 1 5 in the morning the huge I think that you can play on yes OK and so there's something I forgot to say this is not a generously assumptions and it turns out that if you look at the whole element you can identify with With an element is the L identified with the development of the kernel of the formal eventually but you look at the Aston tonics with element of acquiring the formal adjoint it has exactly the same tonics as this is long following you plus so the league so they'll be the leading I again where guess a furious at something when it's a check but the question is proof that probably when the is this if you look at the asymptotic so the cockerel element has a leading terms which has the same form as the with this this this new term over here so this may we might need to adjust the mighty generous assumptions that that's on 0 In the homomorphic case actually yeah you're right we may be due to serve there was there 1 of jurors the things yeah so is an additional generously assumption which probably most people we all used to cover losses of talking about services so additional generously assumptions based on on the this is the only way so it's
possible for furnace assuming that a new clause I'm also said something about the call kernel of views 0 this feeling that something we don't the Brechtian only through the sort of leading asymptotic the recent generic way but I think I'm going to not to explain it .period more distant from the media you're absolutely right and productive suspension on other questions about this you the and there's on the I'm choosing identification in which this thing is identified with 1 it's so so the derivative of the equation that 0 some element the monster of the cloakrooms can call 0 1 so did that just a simplified immigration otherwise I have to serve as anything uses real numbers have to take to the Muskoka moments in his 1 rotation it's not the difficulty level this little cost the title of the book to get the hook it was over come on top of a varied relatives Atlanta's poorly suit too complicated as trying to assume that the boards wouldn't have shadow on you know that the I'm look up now and so next example was a serious as 1 valued case so there can be an index 0 flowing from cute to itself so Europe's index I was gone the reason index I was a and timetables year there's a flowing from view itself so I know I already know that include this criticism analysis suggested that maybe what everyone include 2 copies of the flowing from itself so so tight equals 0 I had this configuration but can say if T is non-zero but close to 0 does there exists a full line which is obtained by including these 3 things together so maybe there is maybe there isn't on Wikinews a similar analysis so in this situation the the 3 going parameters please cheeses as performers Vonage replaced time euros 20 then they did indeed we or 1 . 2 so when material is I'm going to translate everything someone which amended the total translation distance between the upper 2 pieces will be are 1 and the total translation distance between the lower 2 pieces of the Archer so I translate everything eventually by then do the same business was close you plus a new 0 but and the glowing equations available tests and so they're going they're going to be able to glue up to up to 2 elements of the "quotation mark kernel of you 0 so you have like the fate of for here which can it to be 0 Sadie here which is less than a 1 dimensional vector space in the theater there tools and one-dimensional respects they continue to identify this one-dimensional vector space with are and the glowing equations are gonna look like that Sulaiman has you to the minus 11 to minus 4 but 1 each the minus job 1 policy to minus 20 plus show that this is how this will be the same as before last year to the diner plants March 2 the -minus T equals 0 and the he to the miners slender minus part 2 it minus where is the neutrals so where all compressed somewhat land less when the plaster this the answer the largest negative playing in value of the notion that you all the the the plus or minus are determined by absent fathers but you 0 Canada positive and negative ads of his so this is the 2nd
equation sort of is the same kind of thing I had before as the equation you got for this so this is here were looking at the CO kernel of this as a term coming from the asymptotic is the negative asymptotic 0 and its negative lesson taught as measured by the CPI minus on and then this this exponential here comes from the fact that I'm stretching these 2 things apart by distance are too and this could be equation this data is determined by the negative asymptotic then plus history assurance the ball by the positive asymptotic here over here so those equations again and this is just this is a simple equations nakedness analyze them a lot of things to work now and the world a unit of so so the 1st equation comes from this 1st piece this metal piece here the things this millipedes were blowing things the middle peace on both sides and that's why the 2 exponential terms the 2nd equated consumers lower fees are legally 1 thing to it which is why this 1 to general each piece has a glowing equation where for everything that's glued to it there's something involving the asymptotic so that others so Lake 2 If we're going these these 3 top words together so this this chart the good equation for this chalkboard or has a term for the asymptotic said this talk and endoderm term from the asymptotic that the directions from the OK so in this case in this case on it's a bundle over this all pairs are 1 triples are 1 1 2 1 2 and the fiber Over a point is but the cold priorities 0 direct on the cultural the 0 and is now a into pieces which have Coker also for each of those cases you have assignment the company when he gets much worse but I want to Austria since so insulated the abundant example bundles trivial anymore it's actually real interesting thank you bye it these equations maybe is a little confusing how it got to the Vinalhaven down introduced solving this completely elementary so you solve the William equations if it only if all of the above all the following hold so 1st of all the Midas and see better have the same side the fence just looking at the 2nd division the fate of the minor since so just as were sort of fixing the and the question is can they find R 1 and R to solving these equations so the 2nd 1 can be solved if and only if these things have the same size and has a unique virtue they then you have to look at the 1st equation which is a little more confusing I so do I think that that's a lot of that may be he he told me he might reduce the use of human life the value of the way this reminds me of some nightmare had once Blair has amassed nightmare about this ensued when the event and so on so sort of Stratford everything out so that for this this might assign a sort of long disappears so when the fruitiness forget about this was his call a singer like f or something so this is the equation and trying to solve but in the end number identifying this with a cockerel on but everything so it's trivialize the cloakroom identified everything with wide and these are the issues of real numbers determined by the asymptotic I don't helps we at 8 period no idea what the signs of these numbers are depends on the asymptotic so that of the full so basically flaky saying no my in my approaching the critical point from the cider from that side and an instance I was scared away but by that but identification were there at 1 of them is actually clear to positive and the others who had to be made anyway so you can figure this out so on well maybe it maybe it's not necessary to go through the whole thing there's a more conditions for the 2nd for the assault to be able solve the 1st equation so this case is
on the mend when you look at this 1st equation is actually a matter so so it's actually a matter which of these 2 eigenvalues is bigger than the other so lebanese theory the whole thing was going to confuse myself I had to put the work conditions In some cases these depend on the size of on the difference between these 2 values which all that's realistic generic generously assumptions that this difference is not I so it's so the the it's the question of whether or not to conclude depends on the relative size of aided the minors in plus and and also on the side of the disk difference of surviving so psyched for different signs so sort of 16 different cases when can just sort of go through and check which 1 but elementary methods it's it's I worked out all the details on this blog posting the 1 from July to 2014 here early is a worked at the algebraic he tells the analytic details if anybody likes analysis and wants to understand the stuff better it could be a good exercise to do the analytic details also sought to justify all these facts on work it all out outside so we did this in the paper was tossed from work occurs when others were staircase but I should also mention that the thesis search I only of does some call you fold version if this for certain Hall at 1st I haven't actually seen the pieces but to some of the sum related colorful thing by the fragile professor anyway the upshot is we have this configuration in Wantagh will leave readers the question of whether which goodies these to solving some elementary equations involving real numbers problem we can solve them he was so see Mary not
be will solve them found in
general I can tell you what happens so the sort of 3 possible outcomes is just kind of curious so suppose I wanted glue to take copies of the full and computer itself so it's a case 1 is eating glue only fruitcake also warrants on but only 4 are sort of 1 of the possible signs that's the simplest too as you can glue for all K they Forbes said for 1 of the scientists and none for the office they case 3 because he concluded 4 tables 1 for 1 sign I think a calls to for the other side loses his possible outcomes remember we're supposed to get a polynomial which was the 1 policy the customize 1 suitcase 1 you get that well I mean you serve your Circe 1 plus 2 there the case 2 years used to seeing a 1 was steepest he square cut which is the inverse of what was to be the case 3 years you're seeing something like 1 plus squared divided by 1 plus 2 which includes the Mucha coefficients .period was the year getting hurt .period will all this through the 3rd is quite weird but that can happen depending on what the signs are on and you get this on so which of these 3 cases you get may depend on what following trying would fulfill any of last trying group but you always get the same polynomial was important priorities so inside and complied tighten go through all the calculations but they have the base explain the basic idea of how you sort of know tried glue these things and you use it as military equation which you can solve the problem questions about this so I think I have 10 more minutes is that correct was just forget all this concern over so if you look at the total if you got totally lost start over this regrettable this isn't 1 of those things that depend on income from 1 side of the women 1 well so here's that cures the critical point hero here an item spaces for us this is an icon's face of the Hessian and is this is really the smallest possible audience smallest positive I United Hashim the the following you plus generic please continue to come in like this Britain accordingly so coming along as long as this I in space plus some exponentially decaying error terms so those the 2 sides this is a locality usual intuition more serious because usually you you think of a
stable or unstable manifolds which is has dimension bigger than 1 by 1 these agonizes change there's a preferred direction before arriving space line 1 which approach a critical so exactly the generic situation who usually make you want the opposite of more seriously wants to all the agonizing the same makes here is not a generic at all by using the word appreciates what else to do think you have a lot of growth will knew with the idea that discourages 0 because this is famous everywhere and discredit was here the sometimes you know something's truly remarried Oliver prove they just have to work out the prophetic comes out but it's Analomink but just switch gears and talk about the whole market perform only care about something introduced a problem then we'll talk about this had a solid tomorrow so we look at a three-dimensional contact manifolds so why is it 3 manifold Clemson on agenda contact forms Laura what a notary vector field of will to a JOB a suitable generic meaning Saturday satisfying ever-increasing list of desired conditions almost complex structure Poland desirable most of the structure more desired might not be generic but I think this century your desires are you that you have all of this is 1 of the following year the imaginative
but could dating website was looking for a co-leader partner the 300 responses you go to the local this is the right anyway so we satisfy the usual conditions which this is a little of this is a special case of Chris when talk but I want that the armed to the chase sends the turret of the USS directions to the vector field Jason's the contact structure to itself rotating positively with respect to the orientation on a lot of injuries are invariant OK they're looking all over for curves and as special case Chris Wendell and this will also signal is a punctured three-month surface In the end the asymptotic warrants we got best current customers and he had a bomb so these are using kinds of things that 1 must account In the SFT differential in my case I count them in the embedded contact Molly differential knocking the simmering knowledge about either of those things that is what talk about going these things in a particular situation could have yard with here it's sorry the move came they said 1 I think I can put it I could put a bunch of countries in the world was theorems I could put a bunch of years of looking for proves fatal really have so many proofs looking for the here In this OK so you if you so here's here's the problem I have is a problem with that said this to "quotation mark Branch cover other trivial cylinder so trivial cylinder means are crossing for a while so you have homomorphic curve like this so this has index equal to 1 solution one-dimensional model space which means they have to monopoly are translations it's listeners 0 dimension modular space listening so this on to the usual kind of going would be good to have these things I pray like this also reported anyone the remaining break a maybe break along multiple reworded it's like us In any 1 of them not so that's OK incongruities but the problem that can come up so you can the became with these 2 things together and end of the model space of index to curse the problem I can come
up is you can have three-part curve story have it excellent versions of the task you have that index 1 curve on the bottom on so this 1 curve has the simplest case it would have a negative and that gamma squared cleared means that the double cover of some simple really or ,comma the missile Paris I want to positive events both the rave this Christmas and X 1 so it looks like you can glue these because David they have different kinds of answers knowing that put together however you can insert in here the pair of pants which is a two-to-one press cover the cylinder operas can but it turns out that 50 per cent of the time depending on some conditions such as the winter this this this answers index 0 on the general has an excuse 0 to answer sometimes is index 0 so this whole configuration
is indexed to 5 you could say is can whole configuration the glued to an end of the index to watch allotments and the answer is yes it can In his How many ways can you do it on this particular example the answer is 1 way then you can see that by looking at construction went so not a time for today Sodom and due next time is only discuss just does this this simple example all but how do you know how to do this using destruction bundle which innocuous will actually be a nontrivial bundle works not just 2 it's not just the same fire over .period actually interesting on the banned all mention the issues that this raises when you think about the definition of the SFT differential because this means that certainly configurations involving branch covers of trivial cylinders must contribute something to the of 2 differential so interesting to compare will receive a different definition of the to differential next week for it to this so next time I'll talk about how to do this and the genome Thank you thank the question is thank
them he said


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