3/3 Lagrangian Floer cohomology in families

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3/3 Lagrangian Floer cohomology in families
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We will begin with a brief overview of Lagrangian Floer cohomology, in a setting designed to minimise technical difficulties (i.e. no bubbling). Then we will ponder the question of what happens to Floer theory when we vary Lagrangians in families, which we will not require to be Hamiltonian. We will see rigid analytic spaces naturally arise from such families; these spaces are the analogue of complex analytic manifolds over the Novikov field. In order to be faithful to the theme of the conference, we will end by constructing lots of moduli spaces in order to see that Floer complexes give rise to analytic coherent sheaves.
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something involving this time and actually this is going to happen but 1st we have to start with corrections so many 1st say so that's that's cute of the grunting next so is accusing I explained that you can basically build while although some very complex of Q equipped with a local system off the chains of the public and spaced apart from from the base .period to any point you can work with the conclusions of law in this city of star of this with itself and there is a natural laughter from the change in the basement space and Q and will promote a Q Houston play and this is an equivalent this equivalence is that they said last time this and this is just a direct generalization columns flares isomorphism between Gulf local knowledge between ordinary allergy and for the Comanche case but I asserted incorrectly last time that this is true when you pass a completion that it's not true so even if if Q grounds of no more from this it is not true the governor's morphism from completion of that I talked about this time to see if so far have you completely large local systems unless there a lesson the less you read of basically you redefine the morphisms I'm not isn't using save for example the topology on this basis using which you may or may not want latest fitness True Lies this and this was not true and I think at the end of this talk I may give some indication for for why this thing fails but I think we'll be better to just wait until the end OK so now that I paid my debt well you now a lot here there would have been like change here would have a sentimental and here it is not as the others thanks but the casino from now for the rest of this talk on the day or going to set you to be a Taurus again and then adjust unwind some of what we've been doing in that case and hopefully we will use understandings understand things better and especially so that it is not observed that the homology the basement space on the Taurus the images using splendidly Nederlander coefficients uncompleted this is supported in degrees 0 take because there the Taurus the Universal Time cover of the Taurus is already so the only thing that basically exists as the fundamental group it's only human to be nontrivial homered to be group so this is isomorphic to either group rings off the fundamental group at the talks OK but the fundamental group of the Taurus is a brilliant this is the same thing and the group framed 1st Monte on the Taurus with the conventions and for much of the talk it would be convenient for me to just choose a basis each month and in that case you can write this as the Rumple unions would go along colonialism coefficients In Landau and I will just say no and I don't wanna be writing all this all right this has been my notation for this will be slammed me I lost my husband because that's my attention so some the annulment of this range can be expressed as the son who were all found in his urine let's see how I need
to be and annotation is that z today although equal to the quantity of 4 1 0 and 3 the of these are the law models and that's a finance and so on if you know any Andre geometry of supposed to think of this these are functions on algebraic and since I'm not another regulatory I just say I agree I take Glenda minus the 0 to man led Tennessee with functions of this fund rate functions under their watch which notation is that this thing is :colon so now and let's think about that's what this is the guy this is what showed up when we I thought about the exact case but if Q is not exact then we now that we should instead use we should consider instead of this completion which I will write as well and the here put the Rockets cause ,comma was reminds 1 can and when there's this notation me now it I'm going to give the integers unwind are definition all well completions to an element of this spring now again expressible as son of the Alpha nuclear power plant I didn't say here is that of course the elephant climbs in but now there's some additional on conditions this is not the size of their economy of the author by the end I here ,comma we must continue isn't that they say saying this is changing the blues and became a father who wants to this it's supported degrees 0 so he doesn't have any defamation so I think that that that was the skiff let's talk and maybe for 1 segment of the confidently but I think right now is imagined who's walking with the work of the chain level we're going to pass to homology remaining before and we lose some information about the homology but I'm staying in this case we don't but even if we knew was just loses information and see what happens but it's also is in the end see all lies in climbed but as we said now instead of having polynomials we have series so there is a condition and I'm gonna write it now but the Leonard in terms of an expression which I have defined yet as well as often as to infinity undervaluation of sea off on the so there's too on the findings the 1st thing is what do I mean by that it's normal for those to infinity and here I'm just saying you anymore any come on on the Net for example you could just take some of the absolute value of In this the box doesn't matter what you it's but you know that this is 1 of the and because offensive and what is this would only amenities evaluation so so intense and so that to define developed the valuation amount from and thus far 2 2 are can't and what it is is that it takes so the valuation of an expression of this expression of the form they are lying show me to land I'd from 0 to infinity is equal to them this year where I have assumed I paid 0 on 0 and all the other land all eyes were strictly bigger than when the euro In other words this led 0 valuation is the leading exponent From the beginning exponents all bones an element of the so I think that's the definition of that room so this validation of sea off I could have introduced earlier because remember where it is introduced by doctors from the star I can't and I said there amount 2 are let's think about the imprisonment of so the the scent of non-zero elements of another covering so that leading order term has expert and 0 well we already talked about that that's you land that that's the unitary that's the unitary elements we think that any questions so far so L so it's actually you if you May I claim that every element of this completion all the homology the space hazardous forms and you can take that as an exercise that kind of exercise proves that this age has completed because the forest is in fact isomorphic that ended the truth is basically just a matter of rearranging things when we defined the homology groups here we define
them said you know I take up an infinite sums consisting of the
chains and in the space of coefficients around field which is demanding to be like carrying like these a eyes on and then somebody to the lender and what you do In this sort of images collected collect together all the terms which correspond to a given component of the new and that that's how you we express things in OK so we we understood that law all polynomials correspondent functions on land the start to the end and so now the question is what is this what are these functions on so far these expressions is this mother the brain functions on so usually this question might kind of subtle 1 or more if
it is a space it's not obvious which basically is that in this case it's kind of to enter because it's already given to you and something and then some expressions need the correct answer to at the correct way to answer this question and we just need to find we need to figure out which come expressions can on the same climbed into the eyes so that we convergence In other words if we start with given religious no rights meanwhile follow its appearance in lenders continued and we want to know and so so when what what conditions coefficients beyond implying that and that's right after the In the article convergence whenever and while invested in this industry end of Indians is actually pretty straightforward so the 1st thing you observed that is the lies while not in this land started the end in the smaller subsets you land that assistance from the punitive element please then the valuation by definition as we said evaluation of bee illegal to 0 and since the inverse
of eyes series of meetings with leading order term a constant is also a series with leading non-zero constant also series with leading water from a constantly also get evaluation of the ayatollah's so the valuation of the eye of B to the office by that I mean the 1 be off although it could be in the office and also 0 this is just the statement that you and to lender which was described as image evaluation I also have explained last time that itself it you know it's in group under multiplication OK so In particular past equals some of the elephant and the and I plug in methods we have some of them to be some of the neophyte obesity and I want to understand does this expression which live in London does this what no that actually live inland trees doesn't theoretically converge which means can I check the valuation of this expression goes to infinity as well as offer goes to infinity this is easy to see because the valuation of the alpha-beta Alpha is just some of the other guy the other guys here and our assumption was that this valuation systems look at so what we have worked out is but New London to the UN is contained in what you would call the domain of convergence functions pain this the policy minister while insisting that basically some kind of conversion Paris and the exercise this to prove the Contras I if you take .period which is not In the
land of then you can find 1 of these series and find that having here said this expression is evaluation doesn't make sense and yet infinitely many terms our without him with evaluations his on his you get you get infinitely many terms evaluation of the 2 so then you can't add them questions OK so what have we done so we have in place in the world in the center the world of regulatory have a geometric interpretation of these rings and itself the summary after that all this philology the baseless based in Torrance to and that corresponds to algebraic functions Poland on the Taurus and this completion corresponds to well I'm going to say analytic functions all of you let me and so the next thing that I want to do there is also a so-so in particular last time we constructed some modules associated to the grungy and so if allies of the grungy and then this guy and held that I constructed which was a module over this is a coherent shoes all in all I'm to and end and its completion is a commitment as the guy would have incoherent shoes on then on you you that so this is finally looks good you have to remember that against this construction basically only makes sense in the exact case maybe also make sense in the monitor K 7 and really thought about it committee put you down polytechnics and monitor cases that are just put question mark but this 1 makes sense in but if you have to close these kinds of unobstructed assumptions which I was a little bit sloppy about some points but beyond that so in the 2 someone you'd like to say OK well that's all we have that's life but this is unsatisfactory and some selected because you land at Indiana forward that is going to put you on the inside of the store it is very slim and it doesn't it's kind of it's like the unit circle inside as 1 who is basically what I'm talking with constructed a coherent shape the call the system said it's it's like the other than the last 1 and sentence in in the case was constructed analytic complex analytical here chief on this 1 the basically things out of his like an infinitesimally thin analysts living in the stock and the expectations coming from near some trees that you know if you have some situation where there is any sense in which we're cemetery holds many demons in the general case you should still have a coherent sheet or something that's about as big as lenders or something which is locally and about as thick as lamppost I'm so that I'll come back to that well how explained that so we expect To get larger domains Over which are coherent genes are defined this but but but here we have to be careful and so when we constructed but I think the ring of function the model over the ring of functions on on unknown Columbus is the same thing as a coherent chief well that's slightly equivalents but it's not exactly the same thing because Over There were working knowledge category now we odyssey that we're set should be working some kind of analytic category so we should we will produce such such a coherent sheet by local to global the removal procedure was that of trying to build global 1 1 thing that very large bets is all is as large lenders certainly and let's just build something near the unit circle just imagine that we're trying to do build something on the and somehow were having trouble controlling controlling the convergence but we know that near circle we can we can ensure convergence of the 1st thing we do is we try to stick in the circle and if they can do that and you know that we have some kind of non-zero radius around circle where we do the convergence then they'll be another another chart for a year and then we'll get
convergence may be using slightly different model in that shot as well and if you can do this near Everest circle of every radius and you can cover all your entire punctured playing with with such any and producer analytical unit chief on the spot using you using this local to global procedures and that's what's going to happen except what women have time to talk about the details but but the the main the main exception is that instead of working over the complex number we have to work over we have to work overtime so for the next thing I wanted basically justified the the description really given alternative description of these intermediate completion so now we consider lesson plans for simplicity making a point of being contained in each 1 of our Groningen she's going to be a model because being Torres with Oracle 3 signed me 1 2nd and and 4 for convenience I want to make it into the last time which means that but we have a collection we have other flying in a flowing 1 of the Taurus conclusions in and then we have some numbers this commentary on flying in lying in lot and then our politicking is given by the center of .period peak satisfying that the pairing of peace with all 5 times greater than we have today for all mission just imagine that is a triple was it used to the authority it's set a hearing to put that and that's why here it's integral part of the yes some of the sanctions this is sometimes like a technical point so too now what we can do is we can take so we have the inside here fortunate and give myself enough room charges rewriting again he is contained in each 1 of us he Nevada coefficients and number there is this evaluation From land star let us start to operate in a natural mother uncle Monty groups and then so the call evaluation so let's define white to and that the the main really this is the analog of some kind of sickening Of of the real Torres insights he started again and the reason is that I should have said containing the origin the no action is automatic from b it has to be competitive the because it has not complied appointed but it just where I would like for it to so it's it's like a thickening as 1 to the end like in the star in the start again because this as 1 to the end remember correspond to Yolanda and land is the inverse image of the valuation of the evaluation From the verve of 0 and evaluation so that that lives inside you landed when the United States OK so last time I introduced the the last time I introduced on some of the completion of chains in the basement space that is that corresponds to these things than underscored to just right after coming age he the plunder and this is in the in the mutation is denoted by land again convergence series 1 applicant Sr you can describe this as before To as the sets of World Series so you can write it differently so you can well and easiest way to do it is actually limits as the of the sincerity of the valuation of flustered pairing of well-funded in is equal to close affinity for all the yeah these companies others
yes in my yesterday in here and maybe I should make them strictly stressing so that they don't care but the island rather the or not been OK so file this is just again if you if you plug in if you put a P equals 0 you get the condition for that you get the condition you get the condition for being the in well can without without a peace but this is like this this is business as saying the same can you impose the same condition not not only endure at every point along the politics the should be interpreted as employs convergence news and this some of these are this correspondence 2 to the subterranean office of what might have been called a series without peak consisting of functions functions as such acts Best of why convergence for all why living in this White it's like he was this got no because this subset of people also 1st goal knowledgeable 1st commodities declined the and just 2 new features bassist blender started when you look at all the series which happened to converge here when you get this expression which then is this good completed P completed call Waldegrave Shinseki that that never introduced last so this was subsequently had quite he called afternoon domain to their special kind of conclusion the main the notion was introduced by Tate and His this ring of functions is unsurprisingly example the national drink and there it is you take proved that it has good properties for example it's mysterious did so yeah and has invested when when we when we proved if it were just some arbitrary norm then you know as far as I know it's still filled needs so well behaved but I think he loses the Syrian property the same property-based this thing is a comic self-financing many things you can just use these fine at the minute .period to control everything that's happened OK so all I'm trying to say is that while we were times and trying to say is that the name should be suggestive because you have annoyed is basically saying the fees few funds rate geometry is like nomination so too the kind of thing to keep in mind some algebraic geometry is basically you do I find varieties my have idea which break and then Our analytic geometry munitions and rigid and to geometry Over the Novikov Frank them field which should sent is you blew having domains by bye the maps but basically by not preserving their analytic structure end so now just 2 of its time for me to raise board 1 and you will get a yes I don't want to talk about management event with growth in apologies here too I have my was intentionally misleading so and that's that's the theory which you can use to study these spaces and the last thing on the other hand and the less than that I want to call from the last lecture that we can give given Al the grudging we constructed something called any help and had key which is a logical over Over
this spring which we now interprets it can interpret and coherent chief on 1 side last time I explained that this epsilon thing we didn't need we can do something let's less refined which is the thing that we're doing he intuitively the epsilon thing to find some kind of unity Wallach each 1 with respect to some very very complicated norms and while we could to simplify our life and his work with polygon you're not as Nepali carries less information than it's closer to the world of French OK so so you should think of this so this is the cut the the construction of some kind of this local construction so it would be nice as I said before that we have an expectation that we can build she was over bigger bigger spaces but we need a natural source of these pieces they need to come from some natural and some plastic has been the satisfying years I want to buy but because his examples they will be music so let me soon his is the essence of the year the capital the structure and add absolutely everything depends on the whole of what In the change is not an absolute no it's not independent of the changeover to issue the firing what's in their interest you will get you look at it so let let's strike let me try to give an example of a situation in which are you expect to have this largest looking so let me let XV navigate analytically syntactic and let's say that I have in my family that on the other change meditation sorry to ask that the media family a lot of courage in Torrance parametrized bye inside parameters faces the kind of smooth manifolds and let soon many that he there's a condition on the flux remember I define the flexible morphism conversion of flux them in advance of the flight last time and following 2 Commission 1st the natural map From detention space of the at any point until the 1st call knowledge of the fiber at that point and could run that point with coefficient of our or business from office and to this is going to be some strong version of not having a home of that there exists a J so tho the the for all the just to be clear that if the novelty home offices so this this condition may not be but this 1 especially may not be the most natural thing in the world but just impose it this 1 the easiest way to get it means the base of Taurus vibration In the end basically Lagrangian tours vibration and you try to figure out arms you know how
the flocks of those Rongen very has 2 very non-tubular because it's vibration don't intersect and then you can choose to this is actually OK so now madam I think many people including for example like here I have thought about a similar problems but now with but glory families some kind of Hamilton even the coming Tony family it's known that you can build some kind of family from their cohomology which detects something non-trivial information about how the flare series combined be integrated environment but as I said in the New Canaan the 1st 2 minutes of my 1st thought now the telly sending would you get at ah ah our local systems are locally constant OK the 1 reasonable questions to ask what is the analog of family flew ecology and then in the non Hamilton said and there big machine that I described yesterday he's kind of an Allen and how hats and so on and so forth they are the answer incomplete generality at least the local answer on why locally generality that in the generality where the accused old bounding morphing disks that's what these guys were doing not Hamilton logic but if you think about Torah light you can make these things much worse is it because you can if you believe that pondering geometries geometry then you can give them a geometric interpretation in terms of these on the spaces so that we can explain that so assumption 1 allows us to construct state which I will call Weiss said the bye good thing together once for P contained in each 1 all of With In fact it actually quite straightforward because it's basically a version you know I I stay said that some flux not defined the match from the tangent space too To reach 1 but in fact that matters isomorphism then there is the canonical identification so I mean I don't know who this goes back to I would probably say that this is some version of Cardinal Drilling Goldstein although that's the case for a Ford Taurus vibrations and don't be Torres vibrations for it problems so and so In the end assumption 1 is in place ,comma there is the neighborhoods of any points ambient which is canonically identified identified 1 women neighborhoods called 0 in the 1st launched and we should not well if you don't know the about something committed so that the value and I expect that there is such an identification but it wouldn't be economical but in fact it is canonical and this basically the simple reason is defined this user being here is a primer that I will not give you an element of a chopper 1 I'd take this the prime I think of it as views might be used have he's much of the time then I computes the areas of the cylinders that I talked about last hoops continuing so computer areas of cylinders and then get your class but there so review of his neighborhood which is identified with narrative era we might as well assume that the images public budgets shrinking at the names of rustling policy the 1 he said it directions for of the University voice they couldn't .period surveys any that as an example I suspect source I had some sharp eyesight yeah it's not same as the thing that is an example of communal be more just I'm so so that that tells you this amended will assume that that that is a vital part of this is being I have some kind ,comma applying small cooking Colaninno I'm doing this so he gets a cover told the going to us In fact and there is no shame in doing this because this would have to refine his government the contents it is very small he
will let the world know I call on the coefficients the slopes are integral to these quantities are real that's a good question look so off so easily that conference and in fact this is sufficiently following then all intersections all of them call them piece of lines which are intersections and these guys I Wanna Sex and PIK Oregon are integral in fact I O only need double the sections for women about to do it is so now recalled we recall anything like 15 minutes ago the we don't I constructed that it said that there is some kind of half annoyed rain associated to any integral I find .period colleges so now we reject the defined while being to union of all the while PIC modular the equivalence relation this induced by while he Vijay lapping into both the undertaken intersection so maybe you have to think a little bit about why both of these maps are actually kind of lots of of only within basically inclusions there's no they some kind of change recorded the passing from 1 to the other so it's quite straightforward To do this so just as the we like sorry what's going on I'm very little on the board the top recently but amendments so so just as an aside so there is this excellently there breaks is a Taurus vibration and you can make sure the property to hold so the property 1 and no 1 can hold and in addition believes spent the small delicate point man this wise and B it is in fact the mirror the next and on anything at their summit you just ignore the sentence and that's just how it would you forgave could say what we're doing is we're doing Family Circle Monty in the non Hambletonian setting but another thing you could say is 1 of the history of the war has his eye on it won't be In general I believe it was and when the key and whatever the kids and the kids at 9 in the 1st year yes but it's not clear now we've extended its and it's not indicate that lets not with that later made the answer is it's not clear how often it's OK so far and wants to know we would like to we would like a coherent chic 1 white and 1 way to get this we like it ,comma why PRI possibly not this original ,comma that I started drawing on the board and but coherent even the P-I's the dealer just models Over but these kind of ring functions head and then 1 more piece of data if we all we wanted to hear achieved and we only want 1 more piece of an isomorphism I guess I should have given name and into the models from 1 doing :colon let's call them and now guess the amount of their associated to the associated Telegraph "quotation mark models a prominent Shiite penned by supporters of of these achieves Over the double intersections what that means is that new knowledge or a clean and healthy PRI the contends about over this range land that was it when the peak the with the rain associated to PIJ and this should be a softened to the 1 built from Q J so currently I'm lying because of course if you if this will be a construction just a coherent sheet and then when I and number 2 and the talk I will say Well really what you need to do is doing all this in the chain but you can certainly get a coherent chief from it won't be the correct answer it will be some kind of you take some complex and you just take cohomology don't worry about would the extensions of the area of the world but yes we have the model for each 1 of these but the description of how would go from this local thing to a global thing is not as simple as that I'm saying so so here we're working on ,comma when we're working the chronological OK so as I said we already know how to produce these so let me just briefly said when I said that there isn't covered using this this and using this kind of Arnold's classical selected colleges and wanting to recover from its noted Adelson ,comma and maybe even much there may be much much finer notice that this 1 only depends on the support of topology the vibration the 1 about to build will depend on
elsewhere so for all the unit being serious somebody has to tell you that if you were present we knew fuel for the lives common views the subject feasibility of El transferred to I know this is an open convention France resulted in open Commission so we can find neighborhoods such as that of a being this gutless ,comma now such that I feel I need help his transfers to and from the primed for all the prime minister look so you would think that this would be enough this is kind of hope with the transverse then you can defined local Monday between them and then looking very much the same because if you have you know if you're if you have been hearing the prime here and you try to keep track of the intersection between Alan and that the fibers then then you can't you know United's parent transported canonically identified intersections union and intersections here unfortunately for commodities not just defined in terms of intersection it's defined in terms of intersections and the whole Norfolk disks and so this neighborhood which satisfies tragic probably property is not going to be written off instead we have to pick some kind of Jake is really going to be part of a family of parametrized by the interval 0 1 because in that we have some issues with our jails that will for a minute later legislatures high debt and help write something family thought almost complex structures to define to define CS Stars for the ecology all of this the With here just saying you so that the right resulting and then comes this year it was meant as an unjust noted too long to he has very their neuroses missing here I think OK so now comes because of snow class as was decided in principle you would think that there is a very delicate way of trying to compare the flat theories here and at the nearby .period but in we just take the neighborhood policy the neighborhood and being so that for all the primary means that many want to do things like more control the Limoges state like this there exist and if you want them with colored side let's put the the prime here it's a diffeomorphisms of that in the new Finnegan it's a diffeomorphisms is not applied to what was such as fall property the the 1st thing I wanted to make sure that you don't destroy distress resulting so let's just take let's just make sure that this diffeomorphisms takes are Lagrange into its 45 subversive Robbie pride thinking there is this this side beat parts and the prime which preserves are you you accept our grunge and the 1 we have in mind is not hello but its image under this Hamilton interview them which made it transfers the fighter just preserved to the next condition is that he wants to take be too ethnic pride OK and finally if you pull back you Jerry families of Jake and site the prime of pulls back you just write it differently preserves the tamest of each 1 of these 2 conditions are open 1 of largest this condition is open up when these conditions are you can easily see that you can achieve in neighborhood of of the added and it's given such a jury given I'm trying to think and rest of the world yes that that's why that's why the transverse over the entire neighborhood the news over the entire nearby I conducted within the government later but basically rejoinders you just know slide this guy keeping away so 1st you building near neighborhood of the B L they're both transverse Stewart and then use them so when you have then what you can do it you can Canada said fullback J. James N. this is again almost public structure by assumption this can be used to define this flurry of Al PBL sorry waves of I have compiled naturally hold curves for this and hold off the curves and geomorphic carriers are identified by applying society citing but there is mean more of the same but because all the data is compatible with applying prime the 1 used here is the 1 that has pushed forward a pullback government figures Europe before
From your original Jerry just take a solution to the equation you fired a few office and another a solution to the question would now be different under conditions that's all and this is what proves the resultant I explained last time this proves this implies that the not only that an L clearly well now it's called neighborhood peak not that a mail had keys find not only that but that it also it recovers phones and hats OK this is not going to be delicate than is put military here the media and and had at the time and also running out of time but as I explained the last I I said you know what you may say do you take this thing you tend it with the reigning you'll get this but if turns with basically the same ring but over a different man eluded could so but now what we have done is we have can constructive so I think it is going to give myself like minutes on this so we have constructed structure of coherent she Over and acting only cover all something there was this locally was a wise to be and there was this big analytic space Sematech space since the you can ,comma at the base the base my neighborhood satisfying all these conditions for each such thing you produce but you produce the module which we now can think of as a as a coherent chief understaffing owing domain and then the only thing left to do so we must movies together we must compare them and to each other when we restrict To the global and that I don't have any time to say anything about but only to say that it would you should you should keep in mind is not a have likened the 0 here and now I have to be 1 and over this the 0 came at the Republic of overdose but this over this being 0 which shows this Hambletonian diffeomorphisms TV 0 well and is the 1 which uses other Hamilton interview Mofaz interview 1 so the Liberal engines that we are considering over this this this Polygon and the 1 who respond and they're not the same get so when you want to compare over the ball introspection you have to do something but we know what you're supposed to disposed posted a continuation that's what you would you would do continuation of come you can have some kind of moving family of and that would give you a chain of from 1 for complex to the other over every possible point you could build but that's not good enough because he needs to have the equivalent as modules over these big rigs looking so the needs actually make sure that the that the continuation laps converge to meet convergence of continuation OK so you step back and you refine To get so there is some kind of many technical things which I which I decided to overlook and there is only 1 of them which I want to point out which is that in reality when you go through this construction and you try to check whether you get coherent chief or not this refining works perfectly fine and you do get these maps the need to satisfy the could cycle condition and it turns out that bears that satisfying because cycle conditions even I'm an additional constraints which we didn't talk about it all 1 way to see that I know that it's not 1 issue has to do with fines and from images of race and say of 30 seconds so there's 2 issues that are left 1 issue its fine and to resolve the issues of find you assume that's Of these and the other issue has to do with some kind of action which really has to do with choosing a training base .period and in the case of vibrations and we want to assume that there exists a section I haven't thought about what happens in general but looking think but the time only 1 of the year things before we have to this and so if you didn't have the thinness of the almost complex structure like and you can't know or reconsider and said he wanted the myth of the open to so the tenets of the open structure is that he is so here I said you can use the pushed forward to the define home off to define the floor ecology of this thing would be .period look at all that wouldn't work if it wasn't right if you don't have a table most complex after the new can defines so you needed in order to actually compared the flows at the nearby otherwise you don't have to miss the other ways said if you don't have to insisted on convergence the opportunity longer of compactness and if you don't have to go compact you but the series that you write down there she converted yes but the areas changed but I like me to have shown on this this year which along with it was really 1 wants of he said but they Young says he is the broker skeleton all of this


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