1/3 Obstruction Bundle Gluing


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1/3 Obstruction Bundle Gluing
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Hutchings, Michael
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Institut des Hautes Études Scientifiques (IHÉS)
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Obstruction bundle gluing is a method of calculating the number of ways of gluing certain configurations in which transversality fails, but not too badly. We will introduce this technique and show how it works in simple examples from Morse theory, contact homology, and embedded contact homology. (There might not be enough time to cover all of these examples.)
Complex (psychology) Homologie Direction (geometry) Feldtheorie 1 (number) Sheaf (mathematics) Counting Similarity (geometry) Variance Critical point (thermodynamics) Line (geometry) Mereology Functional (mathematics) Number Derivation (linguistics) Goodness of fit Vector field Hausdorff dimension Selectivity (electronic) Right angle Circle Fiber bundle
Point (geometry) Musical ensemble Chain complex State of matter Differential (mechanical device) Multiplication sign Direction (geometry) Theory Number Mathematics Sign (mathematics) Conjugacy class Many-sorted logic Strategy game Bifurcation theory Energy level Circle Continuous function Analytic continuation Position operator Mass flow rate Power series Mathematical analysis Counting Critical point (thermodynamics) Line (geometry) Price index Variable (mathematics) Functional (mathematics) Summation Chain Right angle Coefficient Directed graph
Group action Divisor Multiplication sign Direction (geometry) Sheaf (mathematics) Translation (relic) Complete metric space Distance Frequency Derivation (linguistics) Sign (mathematics) Many-sorted logic Square number Lipschitz continuity Energy level Circle Condition number Physical law Gradient Curve Critical point (thermodynamics) Staff (military) Line (geometry) Mathematical model Price index Incidence algebra Functional (mathematics) Connected space Vector field Order (biology) Equation Arithmetic progression Directed graph
Pulse (signal processing) Existence Group action Master-Gleichung Euler angles State of matter Multiplication sign Correspondence (mathematics) Sheaf (mathematics) Translation (relic) Similarity (geometry) Wave packet Quadratic equation Derivation (linguistics) Mathematics Positional notation Many-sorted logic Term (mathematics) Operator (mathematics) Social class Covering space Addition Theory of relativity Closed set Physical law Curve Mathematical analysis Critical point (thermodynamics) Thermal expansion Line (geometry) Functional (mathematics) Local Group Vector field Equation Right angle Exponentialabbildung Flux
Multiplication sign Equation Translation (relic) Stochastic kernel estimation Condition number Social class
Master-Gleichung Sequel Decision theory Multiplication sign Analytic set Kontraktion <Mathematik> Insertion loss Inverse element Complete metric space Limit (category theory) Equation Orthogonality Physical system
Complex (psychology) State of matter Multiplication sign Sheaf (mathematics) Kontraktion <Mathematik> Water vapor Propositional formula Insertion loss Complete metric space Inverse element Mereology Derivation (linguistics) Mathematics Sign (mathematics) Many-sorted logic Bifurcation theory Negative number Position operator Fiber (mathematics) Social class Compact space Spacetime Closed set Curve Critical point (thermodynamics) Price index Functional (mathematics) Sequence Proof theory Vector space Order (biology) Equation Normal (geometry) Point (geometry) Inverse function Civil engineering Student's t-test Number Pi Term (mathematics) Operator (mathematics) Theorem Stochastic kernel estimation Subtraction Condition number Module (mathematics) Addition Standard deviation Uniqueness quantification Expression Physical law Mathematical analysis Algebraic structure Independence (probability theory) Set (mathematics) Line (geometry) Calculation Vector field Universe (mathematics) Family
the command thanks to the organizers of the the right so anyway so supposed to talk about obstruction of and the references are so that these 2 unreadable papers by myself and togs good thing something something something Parts 1 and 2 and the funny part begins in the 2nd paper Section 5 of the explains the gory details for construction going for Holdorf occurs in 4 dimensions the ones about ominous start with the simplest nontrivial example I could think of witches from more that examples of my blog floor homology that were present ,comma is posting from July last year then I'm not going to talk about this at all but there is another nice examples similar techniques in the paper about retained by incoherence my contact knowledge 14 12 . 0 2 7 6 in my friend had interesting also so what is obstruction of doing so and destruction bundle Gooding where gluing things went salary fails doesn't fail catastrophically so you don't make the Polly falls but it kind of feels they need something directions it will be interesting later to compare this with the powerful so to helmet next week is going to talk about the definition of sobriety field theory by Polly Foltz said the powerful steroid get some numbers these numbers satisfies some properties so that a unified variance 1 of the many don't know much about these numbers but in some situations you actually want to know what these numbers are you won't know what's going on for example if you have you have the situation where you know it all home off a curse which the tranches only fails to honor 1 of the numbers of which accounting these things the obstruction bundled relevant to this question is will see OK so here's the tricky here some Mauceri example one-stop it's it's it's easier than all talk about the whole Norfolk later 2 I want consider circle Mauceri a circle value just to this phenomenon that I wanted to tell you about 2 per cent have 5 internationals manifold then we have a more selection and you choose a metric find agreed vector field but and social mores means that if have a certain function locally it's derivative is the director of a real real-life functions so locally looks like a real life so the same definition of what's worse functions it on that and you can define a Amoss complex where you do the usual thing you count count creating full lines between critical points the only if we have to deal with careful because you can actually be infinitely many full lines between any 2 critical points so drove to fall back to sleep and is circle direction the picture there is a critical point over here on the floor and some critical point over there before lines can around in a circle
direction many times so that can we have infinitely many of them said to get a fine accounts you pick some level said that doesn't contain any critical points this is a critical point the the theory Final Four Nova covering which in this case power series for variable With coefficients and the differential the critical points apiece in next critical points the sum over technical points skewed and excitement but then we have the sun from calls 0 2 Trinity the former variable To the decay some counts flow lines from PDQ came here these are the source of thing the across the levels at the Cape Times the following that played this is countered with you know 2 0 with goes crosses 3 times it can and you once you once you do that you find accounts that so the circle more serious confined in ways which are completely irrelevant to my talk for now they say well Is this invariant undivided do home atop the of the function if I change the metric to replace it with some of rhetoric calling at the same quality and the answer is yes you can prove by following the usual strategy of defining continuation maps but for certain obscure purposes which were beyond the scope of this course sometimes you wanna know in more detail what the what the continuation chain that actually as you to find an explicit 2 1 0 sort of what you want sort of deformed effigy home atop the Navy sometimes during this whole atop the fail to the generic when you want to know as you cross that non-generic time what happened what exactly happens to the change the state against the yen so have a this yeah I think we can do that although police still have to count the number coverage because you haven't on compact what is this you have to see why it is it's that simple tonight have pulled it's not at times equals 0 on and the various ways it can fail to the generic for what I'm interested in is wearing have a critical point but the index I say the
floor in front of you to itself "quotation mark chemistry fracturing of what you you 0 so full of computer resulting so this the reason why do encircled by more serious for this to be possible again this can happen in real life worst encircled by more serious to happen what was supposed at the center sexy just once no 1 wants what going happened to to the chain complex so you can have some other critical point key the next item plus 1 Click here then there could be a full line was called New clause From PDQ on so this and so this this thing has index equals 1 this time around there it's sort of fun unwrapping the circle specific suggested to draw the picture we can have a bunch of copies Of this low-lying 0 from here itself so this is a here's the fiercest pretty non-generic thing you know a broken thing with many levels and then if I perturbed the little away from 0 what's going happen to them some so you knew it could happen that petite slightly negative personally positive there's there's a full line like this chamber including all this together so the question How many wastes include for a team that was prominence that's can tell us how the chain complex changes affairs maybe there's maybe there's no full I'm here when is negative and there is certainly a full line of years when to use positive so that the differential of is getting added to it and to teach them something changed so this is why this is a simple simplest example of a problem I could think of for which we wanted to obstruction but bands I sort of know what the answer is for obscure reasons so tell you what the answer is that they were willing to try actually understand what's going on so this 11 what is that there exists a power series the equals 1 plus or minus 2 and then I don't know why we know such that so the differential after a slight where would slowly positive City plus interest the differential for 2 only positive the differential for use negative are chained by conjugation by about ECU ECU Q In this power series in 8 times here and in here it's all other critical points is basically is account so full lines that are created destroyed this bifurcation on and on it's it's a fact which follows by combining various obscure forgotten that that in fact this power series is 1 plus revised seated plus minus 1 4 possibilities the signs are not related I want understands directly by the destruction of any the heart of what was going on and what the people in the mind of the Office yes organized at this depends on some very subtle but the nice thing about the destruction of doing his analysis will if we look at the analysis carefully tells exactly what's happening is very precise this but we have to look at it carefully the all of "quotation mark yeah right it was called here nobody's used fuel OK well if I were going to have to learn to tolerate something bigger what were whenever it but the thing the capital to have that center yeah In case sovereignty
Bloomberg this is not what the law this this summer yeah so this level but it's proved spot during Monday's suggested and through unwrapping circle and looking what happens there other questions about this this is not straightforward so the earthquake for 5 different signs and put them all together at his house was owned by the way from now on source To avoid my had completely exploding I'm going to replace them With seem monks to confessions so normally get account lines was signs but I think it's sort of indecent discussed in public and so so that this I know under matters this size still matters so I said what have geological versions a is either 1 policies for blind plus 2 plus 2 square was but How do we well so it was just to warm up before oriented book what we don't have about you think here the of freedom and the hanging onto the boards and like lifted up so "quotation mark from so warm let's let's look at the need to prove it is equals 0 so this is something which is supposed be easy From to make it difficult but so I mean review this in a slightly strange way because the destruction bundling set looks little different from the way people usually going but some dealers in strange ways which is going generalize conveniently it's not transfer situations this so we have a flowing and access by minors in disciples flowing so you want circuitous index I met another full line you minus 2 critics .period -minus I wanted to do these things To get and of the model the person you are so these what's right for on gradient of staff think of these the opera gradient in these these awful and you satisfy the equation DSU plus the interview equals 0 some thinking I really should try the Arabs from the West to the Press that before the full and actually go up and down the use but yeah I wonder who these things also the so Thursday on start last year besides killing effect Britain also the a linear station of this equation which already has the so this goes from sections of the sir you start yet well and we probably won't use some modest this completion it doesn't really matter which 1 as long as you have to be conditions for the L 2 1 sections so this is a derivative of this equation with respect to the defamation if you do you know see it is that could very derivative I see the attraction of and minus the cauldron derivative the vector field be in the direction csce factors that allegedly connections so that's snaps having these things so that I can choose a very large value I'm going to translate you was up and you might down totaled translation distance we it's called police things apart the distance or an animal has some kind of functions so as you close up by applied by Ava to the United States what sort of both for periods USA maps defined an hour someone composed that with the translation of but you need to choose from ,comma functions so ,comma functions the minus which look like so if you are over to his that's 1 of the here somewhere between 0 and or a 2 2 tie the name of the kind of function plants like that the poll is progress and the size of the duration of this kind of functions and the order of 1 of the hour the bomb and then what's now but less so basically you're pulling apart 2 of them so that jury at times as equals 0 the closest to the critic and close the critical point Q the course charted by next week we can create these things so this is a curve which appears on
following up the incident so it's been minus times you might as well the last time you so for for for less than Asaba to it's just you minus as the are
2 it's just you plus and in between we use this kind of functions to interpolate between United and plus but the usual approach would be start with this thing that the perturbed you argue that you can paternity and an actual fall plus for shifted the news the intention was I can't go to the already even shifted so when it was such a visit translate them up and down didn't think using notation for that to this is these are the shifted the translator sir no I'm not doing anything other than the the biggest too dependent on yes moving ahead with yet the same kind of thing so in the picture if you want a picture that the head of the church and you put plug in your life the awful and so has increased so 1st were following you minus because you might and translated were following for a very long time so very close to him and then these ,comma functions curricula and we start interpolated into class and following us like that I had a quarter-century years all the Carter functions toughest taking place inside this quarter charts of existence the but that's increases so were following you minus the distillery close to queue and then there's the risk ,comma functions kick and were interpolated plus the following year the public a of I don't want to do that I would love to see that this right yes so to this edition is defined with respect to a quarter charge in which the critical point corresponds to 0 if want to say this little more correctly I was talking exponential maps and so on and trying to make some calls possible OK so that's a pretty good curve and the usual the usual way that's is presented used to say you know trying you know argued this can be perturbed to natural for them to do something a little different in which I'm going to perturbed the things before a group before previewing and then pretty good so someone like this so Oregon so miniature Her let Cyprus reminders small actions sees of small sections pull tension and we're going to look at a curved so is considerable the miners times you might as "quotation mark plus the times you why services so this is my free good curve its modified using sections of the Will modify both numerous new What's so I knew my this is returned by sigh minus but then here you plus this is perturbed by his was in the middle East for church fires from some combination of but have under the law yes plus numerous voices but I want see whether I want us all solved for this to be a solved so when this fall so right
but this on the sizes 0 in the waterfall I want use the size of a beautiful songs often when right down the equation for this to be a full and salt this is a little messy but this is really the very simple as case I could think of of but also led to make life even easier was assumed that was in the Islandia in this corner I don't have to accept terms in general but as I wrote these glowing papers was tops and services like my analysis training 24 and is basically you sort of write-down equation and that is the stuff you want a hold to crap you need to have to concentrate on practitioner show that doesn't matter that the Fed tightening them look trying to make this so there's little crop as possible at the Salesforce rate equation so I have Diaz might be of this thing the minus 10 to minus plus I might have lost the plot turns you plus plus I plus so what is this sir there was a D B minus times you minus-4 Simitis that was the A-minus times the promissory writing cover interest that this canal where about 70 s you might with the -minus there is a similar thing with aplomb but if you don't mind I'm going to take the liberty of multiplying this term by paying a plus and terminated by which I can do because the support of the of Baden-Baden miners is contained in the region were bitter plus equal 1 section general attitude doesn't change anything that I have to put the vector field so I have my mind 2 the mine "quotation mark the derivatives the interaction but and then in general is going to be some additional return right Cumani was assignments but basically this is the Taylor expansion of vector fields was the vector field was at 1st a pulsing quadratic terms -minus state of flux and if you viewpoint he knew the origin this is all this only comes up with away from the outside of a quarter you would think that the you just so-so this to be interpret as to be interpreted as the exponential map of you minus evaluated on Simon so I'm sorry for my supplementation just tried look at how we can cancel some stuff out because is so you minus is a full line to DSU minus was the view minus equals 0 across the South the likewise you pluses of aligned so I can cancel this TSU plants With this reply and when I get so you get the beta minus time so when the nice thing this term relation acoustical few of the a little more come so this now has society -minus Simitis the is the at defamation operator applied to some these 2 terms here D minus assignment this is the minuses the definition of Wendeline rise to question for you but then what else is there so Houston life and that the state applause times you want .period and analysis in terms of the plot times the
plus side lost plus 2 plus 2 plus 2 plus 2 minus these cues to craft on the Net this type of humor so this this thing it is the failure of my thing to be a full to Wannamaker thing equals 0 the more general the more general situations like in the papers with served as a bunch of extra crap in here but you have to estimate With stock or perhaps 20 minutes at a place where we can actually have
a summary right this -minus times the minus side minus plus plus but think they have passed since the policy was so my equation for thing to the fall on the 1st landing you need is that is for a sufficiently large Darren there exists unique so I might as well such that because these bonds these are we have to strengthen the conditions they don't want to a unique here Simitis class the sight of her mind it is perpendicular to the kernel of depots minus so is perpendicular and the L 2 cents and I'm assuming that where situation to fly to these things are cut transversely so situations D plus or minus is rejected and kernels one-dimensional could get given by the Director of the translation of the full so is its unique pair all such that both the demise of the and
now if we have a lot of them equal because both of the theater 0
then certainly the summer's 0 and so to fall the idea of approved some not going into
analytic data because it'll
be too messy and take you longer to be a little more honest because I don't remember all the stuff from the paper which was written years ago he 1 of them the reason why the invention of writers I would say such a milestone in human culture that he no longer have to keep things in your head you can break them down and forget that anyway did he have to prove it's too but firstly rate equations select this is what worry that were what's limitations on M so that I was like system would not actually on the steps of the during the day and when insert these are decaying as you go to the end of this notice here also so you don't you know keep this quarter inch screen losers here In so deeply minus interest which is going to go to the target is your favorite favorite artist is completion of the United Workers the on orthogonal complement of the colonel the deepest but someone rewrite this equation the equation so what is there but look at the equation the miners who was here to this that D minus sigh this people to queue outside my plans to spin applies to you was like so complied in that's equivalent to saying that some members assume Simitis orthogonal the Colonel this is the time minus is equal to the miners inverse for use by the way of you want to fight likewise the pluses equals 0 the only side plus sequel to deport Q was signed last Tuesday 1 so this is the decision the equations we need to solve if you want to use the contraction
mapping theorem in your favorite past this completion so you can pick your favorite promised space completion and you want to show them that the right side of this equation is a contraction mapping on the set of Paris Simitis plus you look at the Buddhist this this term of the EU plus a new miners is a constant so that's fine but I have the derivative the Tyneside was the derivative of the classes on the order of 1 of and I don't know what the operated normally do you mind versus but if I choose or are to be much larger then the operator norm of 2 minds verse the that is part of the equation the contraction and also this quadratic term if you set things up so that sigh minus I see 0 small business in this part will also be a contraction on the the solution such an offer is good and then yeah the facts to check some of this construction gives you but but we from the sale of or a sufficiently large to set of so if you choose to different parts India to friends we 1st as well and any any curve which sort of close in sense the usual compactness to this broken curve is actually obtained by the construction it it's it's a slightly weird construction because you look at this thing here so if you look at time minus then where else is large but wake up up here in the picture please sign minus doesn't matter at all so that sewage owing to the curb appeal does independents and minor civil law doing to the curve down there does independent less the still but on solving 1st Simitis inside plants which are defined in the whole it's a little weird because this I might inside plus contain various additional information which I don't care about it's a little weird but that will see this as a useful way to do things because it allows us to use the analysis of these operators to cross into 1 yeah where meanwhile the latest using this equations I have this I had evaluated on this whole expression and standard of nearly which I couldn't do fewer nonlinear but if it's not linear than just an extra returns it's not question I don't know what to say there this unique Simonson supplies satisfied conditions they didn't put all always conditions on it will be unique in the world confusing vocational student example where the structure I think the is I think we will but he can change their minds since I lost sold in the region where they were both the Code of functions and 0 you could add something Simitis and subtract something inside of me at the same time however they no longer satisfied equations the demise of the possible here's world number 2 any more interesting world number 2 and I have a 1 friend family functions and at times to equals 0 but by there is a full line of the index 0 so this goes from future to these both in the book and then there is another full of critical point lead for next I was and then I wanted I wanted glue these 2 the actual full from computer are pretty article the thing is you actually if you can do this only for 1 sign of teachers either you can do it pretty positive and offer negative revise 1st the analysis is actually tell which and so how does this work so what's the cans with the construction of the again beaches are very large which translates as before father and then what's the data the end the United States and Japan and so this unassuming on this call "quotation mark call for day on we want you the while I was there was different in this situation is that there the operate Afridi 0 is no longer subjective as a one-dimensional conference then we we can't lose these for fixing to we have to change the world so the inputs for the glowing construction Our assignment since I was 4 plus plus 2 someone change through different small-time too temperature by Simitis and and then from these inputs we get more and most that is 1 thing to do with when I understood as an example of the growing construction the nephew on the understands how Moss complexes changing bifurcation so this is part of what you have to do OK so than this you get a full line only the Gambia minus 4 the miners tires the minus side by side close to plus the times they want there's no money is 0 of this this can be induced similar calculation to this this could be good the index returned because when you change too you change the vector field some so the status here I side plus 2 is the quantities here last year the qualities here a size 0 this can be a DSP apply you policy was sigh plus additional terms which writers to tee times the price for the pride the derivative of the vector field suspected and then there may be some additional quota terms which according to a similar expression for the so the applause I lost my do as it is here you I here was too tentative the question is published he said meanwhile don't you want you could say that the rest of ministers said that shortly so here this inverse operator was defined the whole targets based general must be defined on the image of the offer no well it will work up to bear a child that 2 years when he was placed on the way to discuss of with the them 2 of them the small there are still a lot there is a unique here Scioscia serious plus such that before I 0 perpendicular to the user so was is perpendicular to the kernel of the class the loss of sighs heroes .period 2 sequences here and as for the 0 I can't necessarily get equals 0 what I do all I can get a cat this perpendicular to the image of In the years approved Proposition during the year the proof In the contraction I to endure that the universe operators only going to be defined in the image following we wish it absolutely applies here he protection to the end of the year I said then and that we have a D 0 inverse grows from the image of the I thought complimented her on Tuesday Oregon solving equations that looks like what we want pie 0 the heroes acknowledges that she was on so I can write this down hi 0 on d 0 5 he calls what will that can write this such an the opportunities during the 1st 2 sections of the society here on the coast inverse global book but it was a pleasure here 2 0 interest criteria plot of what they can do the contraction mapping the before what's the punch line so quite didn't quite get as far as expecting to get and some of the Community of completely explained his temple today but I tell you what happened said To who I really want fate plus in theaters 0 to both equals 0 and I can't get there all like biting at data plus 2 equals 0 with a 2 0 0 it is lives in a one-dimensional vector space a compliment in the Colonel 2 0 with one-dimensional vectors because they can't get that was 0 so now we have this picture over here sir I have all been set of all appears to be that over this I have an destruction and that this is trivial or fiber over any point is coke I have 0 this section what is the section of of succeed ,comma are is equal to the state is 0 it's a good instructions I can almost get going to work is that the state is 0 my God equals 0 misstated zeros the obstruction to good so when this is equal to 0 I can actually glue think so but this is some function of tea ominous functions 0 identical finally run out of time so next class I'll tell you had actually compute what his function and how this figure out whether you can move for teenagers Richard positive around and then to the harder example hold 1st ball tried all try to make stuff modular like you know the Titanic has compartments of the 1 part of it fills with water that is sealed off the the ship would think some of which I do that so she didn't understand anything else as long as it to make EPA's World Water Resources Inc 1st of all be hanging around and we want to talk about this stuff but I have no idea I'll be hanging around after all of my talks answering questions "quotation mark office this is depending on where you are going to other examples of the last review for this 1 try to make this 1 think b


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