An Integral lift of contact homology

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An Integral lift of contact homology
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Cylindrical contact homology is arguably one of the more notorious Floer-theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations has tarnished its claim to being a well-defined contact invariant. However, jointly with Hutchings we have managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of non-equivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to SH^+ under suitable assumptions, though it does not require a filling of the contact manifold. By making use of family Floer theory we obtain an S^1-equivariant theory defined over Z coefficients, which when tensored with Q yields cylindrical contact homology, now with the guarantee of well-definedness and invariance.
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and as the you can thanks for the introduction and like to thank the organizers for allowing you to
talk to the fact that case of a lot of us will be joining with Michael and I was hopeful he came with his obstruction bungalow in an intersection theory from the Ch he was kind enough to lend them to the contrary to my knowledge are it's Oregon started off with the context manifold on promotes the target will be three-dimensional would in some parts and Mike get the play about what we can do environment and on this is that maybe not degenerates the contacts and it's going to be cold oriented meaning that there is a global 1 form that defines the content structure and um for grading reasons let's pretend assumed that the 1st first-term class of the kind that distribution vanishes and the I'm Not In Our is going to be my married vector field and then I'm going to have a J it is our almost complex structure on fire Cross and In the deeds the child Linda so here towers then maybe like ordered on are maybe translation environment here translation invariance didn't our are is going gave the panel medieval to are on and they once did they are restricted we're on the same day as the the windows compared to forget that we want to arm Council undergo Kurds armed so Christopher and asymptotic leaves little Kurds are the domains is going to be a cylinder so that I can be super exciting but they're still fun to look at clause in all those days of fall the lads Gamal plus gamma minus the periodic framework of it's on and you Gamal was gamble minus the news of Javal Norfolk curves from a cylinder our process 1 g 0 1 into the sink like vacation on equipped with this view and OMB are invariant almost complex structure of and then we once come to you and jails in the region and the use of London's home fixed and now we want the limits of we take on odd to have of cheek as 1 hand unescorted we want the limit as US goes of course minus infinity of the rejection of you and the contents manifold component to be every dramatization of on the raid were very at on Gallup plus if we got a positive infinity again minus we got of minus infinity and then we want the limit is as close to foster minus infinity tub of projections they are component of you to believe Wasserman's infinity so on the 1 I think like if my current manifold was something stupid like a circle that the accounting firms like that you will assume minus on and then there's also going to be an our action on the technically you have to modeled by on coming from translations on the on targets from and so From we got a look at the differential there will be translation actions but a lot of times all ignore it when it comes to talking about index calculations and a gave sold on this that through the legislation on no but I just felt that he had history yeah I mean eventually have to the 18 drop and uses it to reading of course defined ,comma dynamically convex context formulae some sort of way of making sense of on degrading and intractable the arms of the Grameen was given by the colleagues in the index managing complex formula we have that the virtual dimension of this model is based is given by Gamal last when the green of gamma miners in here the greediness an rating so this is going to be a seen year of gamma plus and minus 3 so if the dimension of our contact man for all scrutiny than you get the suggestions you gamma minus 4 so that I can tell you what a chain complexes so this is going to be on generated on guy only good reworded so we think this set of all brave orbits and withdrawal every word it's because the band all and so defined everywhere ritual annoying but soldiers the CV acts of a framework on Canada I on the same period the iteration of your orbit Oregon flip-flop between being even on sorry I hadn't even if it flip-flops menus throughout the evening multiple covers so Battery Wharf it's you don't have I talk to the Internet you want a real definition on everyone
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intervention 3 federal workers are on you have multiple covers negative hyperbolic I mean that they have on the iron in value Of the linear eyes return map if we restricted to the kind that distribution is negative and real perhaps the reason for the year even through which the deal In there is a lot of is anatomy this is the vector space generated by all the reworded neutron the battery work at his madam you could be on the history of the life of a soldier keep managing complex even if you had all the trends resulting the world you never revealed the proven variants on glitzy see release on days you can't Orient's my on curves that landed on bad reworded its unless you have some sort of a parameterization using of your very worried and sold there is when you try to sign orientations you can get both positive and negative orientations assigned to the same cylinder and this causes a headache and dumb In order full Mauceri can kind of thing of other battery were events as being like the bad points whose like reading full lines get reversed arteries whatever action your money now bye many other questions 1 of the yes that this kind of incorporated by the fact that this is the projection onto the on contact manifold is a repair registration gimmick was minds so that already includes AT & the differential so there is actually 2 on equivalent ways over to on defining the current differential of so either we can define it from it says so in the interest rate the government analysts OK so either you can define it by looking at the multiplicity of the Tovrea were you were you can define its by including the multiplicity of the bottom right border also so and Absalon is this was the centuries of coherent orientations and my view is that on degree of from covering up of the curve if curve was not found somewhere injected a multiplicity of your a word of it just means arm you've heads iterate some simple some number of times in multiplicity will be the number of times he reiterated a simple reworded to get access and so on it might look like the use of qualifications are defined or not integral but artifacts integral on and so you can see that this if you have a 2 1 fold cover you have axes gamma the closest to the KP and why His gamma minus the kg this would be a clear 1 fold cover cylinder arm down plus to the pedia them online at to the and so the multiplicity of axes KP the multiplicity of wise kg the multiplicity of this curve scale and KIO survives both Cape Fear if can 1 theory history because the and of the accident was 1 facts is a symbol reworded also may be symbol hasn't been defined but that just means that the amount taxes on are my seizing into kind segment of wasn't attracted tens of armed and of you equals 1 if you lose so weird injected so that looks like a beautiful theory what is not really so beautiful I so some of the issues that are going to rise is that we're working with multiple covers the question for 1 goes the multiplicity of the 3 were 1 includes multiple see the bottom reworded your Song it turns out that the equivalent service he had all the chances telling the world that we knew differentials refined you actually a homology and then on and a chain model you could just divide and multiply out by multiple suitors so everywhere it's a pass from 1 chain complex to the election comes you know if you have a magical transfer herself the yes but the equivalent so was very possibly not so correct but it still that 1st issues that were working with most covered current interest expense money lectures telling and as well as Jews and others ideas about the horrors of mostly cover turned and then also I think Michael mentioned this a few times but you can look at mostly covered curves Lincoln and
of having on negative index even know they of them evenly have to be there and you can just use of generic provision to get rid of them and listening like compactness accuser Osaka ,comma going on softening the beautiful on what they make the theory Carter all so everyone's favorite and year-round from 2000 which is isn't of the variety Astra argument on hold says that soon today Our communal contemptible I really worried indexed minus 1 0 1 so this is a list of grading that's the ,comma and expose monastery farm and although also started by saying that while I the I need degenerate context manifold the differential and well-defined 1st normally Sunday which differentially reducing which heads mother aware of confusion but that's OK it's congestion Eric came of the U.S. Circuit of homology so GQ where 0 and cosmology which depended on from 4 year contract form the trace of was actually an invariant Of all the content distribution so as to reduce dependence Of the choice of land and almost complex structured in a by which we will be in place in the history of some of the provisions of the the new uniforms he feels that this is the part of the world this is the widening appear in this form in the paper appears as a series of white 1 is the proposition that the universe is also in the revisions will at the beginning and said that this is all a conjectural while Watts was Leipzig on conjecture calls the lecture while the other problem is that some people there said you don't need a provisions was works anyway there was happy no 1 wanted it could have be sold under this beautiful compactness for alluded to this is bridge yes recover that contained the result from 2003 again I'll paraphrase this result so there no globally all played in Hamilton in Florida married but we do I have long breaking of the Kurds into the buildings so-called Michael drew some beautiful buildings of suitable Maupassant category cylindrical Civil records on and so the heuristic reason for why we have this breaking news because I'm so there is a maximum mom principle In part components of the selectors Asian birds you know can grow growing minimums and so the dramatic impact of cyberspace on you're going to be allowed to develop new punches above the on bottom and so did so in the next few Saunders you could conceivably started to have some of the picture people draws you have some sort of a minimum developing in an hour direction insulin if you want a compact fighters based on could pop off the plane and on this would be allowable elements in compacted case and if you cannot serve excluded for other reasons and then this makes said broken soldiers in you're competitive side mightily states and extend its private life was returning home the National Assembly of the problems that resulted in the history of the amount involved and In the Still of had a dry complicated sequence of buildings but you can imagine that sort of Michael was talking about how you can have on branch covers trivial Saunders those guys can you know have index 0 civilians in some situations where the axes gamma squared his gamma and then the cylinder breaking the bond demonstrated Gamma Gamma gamma and this could be a whole Norfolk .period index
to on this kind of index 0 in the sky his index 0 so severe Walton rectification and in dimension 3 is really nice on iterations formulas for the colleagues in the index was on Americans Rosselli soul explained in a minute you can salvage this theory mentioned during the on button higher dimensions you can have long cylinders of index 1 you take the most Culver and they suddenly have negative index and then that means that if you In other elements of your compact aside marginalize space are going to have to include things like a chain of Saunders where you just have to and about being able to edit to within the soul that makes things complicated and the funeral philosophy with a lot of this stuff is well if we only had
some injective curves is Vijay generically so all of my modulus spaces have to have lot positive on virtual dimension because working in support positions have are action and the only way that you can add 2 2 with positive numbers is 1 plus 1 on this original assumption about no contrived Florida minus 1 0 1 to come because all the way to do this sort of building is not excluded by the assumptions flight on armed you could have some building where you have 1 to 1 and 1 services with 1 eye on the assumptions of Kenjiro exists and then on the 0 minus 1 sort of come up when you try to prove on invariants in homotopy depending on what level changes so that you want to say so I have the so it turns out that armed intervention 3 you can arm gets home automatic transfer Selby and the index calculations work favorably and so can make is a founder of not very a large class of Kontic forums for the class of some kind forms rejection find kinds entomology arm so some sort of the definition I came up with the CIA contact for this is dynamically separated on provided the companies in the index again is between 3 and 5 4 Gamez .period intractable and every time reiterates gamma you're just increasing the colonizing the index by 4 months of this holds and then plus you can do this is you the decision saying gamma because none intractable have analogous all definition but you have to keep track of Of the 3 homotopy classes of reworded its images becomes a master at found the question in France but the other and then there that approach arm Is that it is In this so this condition this is usually curable up to large actions and action because there in a role on along damn office reworded so wary of 2 large action I mean for a whole reworded serve on action less than regal something you have those dynamically separated property and these guys all home for the most part arises also is a natural Class a provisions you can put on three-pointers decisions spaces so this is really these
guys come from so if and is why 3 plan is meant generate many of separated Ben NJ generic on those wonderful context analogy is defined on the sometimes ought to wake some college action that sort of thing so if you don't have those dynamically separated arm for all of Europe were minutes but only after some large ejection and you can only find political context knowledge of Tibet larger action and then there is a way to take direct women's sexual sort of computer home variants on under the choice of Jaipur and dynamically separated of fatal this is not so because it excludes a lot of examples like dynamically on wake Sunday in which backs things like ellipsoid so up with someone that knows more than you can then ethereal what thing is going looking work so on a dynamically convexity transiting therefore there's something that over that stands in the studies in the nineties and would catch renewal as studies the papers when she was a graduate student of so I think dynamically so 3 on his claim and Glaxo provided the 1st Chern classes of the content distribution restricted applied to wide is 0 and the companies in the index of demonstrated regulatory for all contractual warrants became almonds over an example is the enemy strictly convex favor In her for With the standard syntactic structure of which transverse to the radio field so this is a nice on classic contact manifold because I raised the arms but it's basically without answering the theory of abstract provisions it's going to be the class of Kontic manifolds on so that all the contractible Ray it's give rise to planes that will have to have our own version of the mansion at least 2 so this would be sort of like a nice glass of manifold you'd like to prove our had sex knowledge on is defined foreign variance in the dimension 3 of the used what you wear Water Mill intractable salt Lake condition a C 1 of pilots 0 or this so what Michael and I proved is that dominated wide window is dynamically comebacks generic then the only thing the building of the index 2 in part crossed by Justice cylinder but once broken cylinder for all but a few articles on assumed that our colleagues in the index of Uganda's strictly greater than 3 4 gamma nonsense .period intractable and I'll explain why in just a 2nd or you have a pair of pants of plane and materials under gamma square in on the euro but the story of the century assumption on their arms which began the 1 of its assumption advertising the index's strictly greater than 3 for dominance symbol and intractable on respected is removal but we have to do a little bit more estimating things to make sure that something that doesn't happen in the next blackboard on what sort of Will we proved is that the only building the index still far either an unbroken sauna once broken cylinder or this pair of pants configuration on if you assume if you don't have this assumption that you're a pair of pants configuration on can have like this damaging the damage of the do want us could be damaging the G 2 and this is just a D 1 was due to hold a retrieval cylinder and the 2nd part of alumni that so no index to can limits to the on 3rd configurations so we didn't prove that there are no buildings of this type there certainly are on low wages showed that there is no index to cylinder that could limit to it on which means that has a corollary armed those missile into account technology differential is well-defined and squares to 0 this is In isn't decision on so the reason why we don't have any variance yet is because this sort of analogous 1 border zones for sure does not hold any questions I don't know what possible with all quality variations on the part of the plan will be this it still there he's still there this will during the meeting he will get
some or all of Europe and actually do that together with other than what you needed to solution for you if you like what kind of perturbed and you want to do so if you're doing obstruction on online and you decided to try and see if you could do this number ways glued together 0 this is the 1st known who you think it is approved by contradictions
are so used to yeah you use intersection theories say like suppose you have this problem into those then you look at some intersections theory properties in approving a contradiction there for the index to Saunder cantonment of the winner of that type of computer rooms you can see the Yeltsin's obstruction 1 goes back to Winding numbers and on the what can you do it another way the only thing I can say that this was going on you know the 2 of you have In the 3rd time this has 2 sections all intersection various office to prove that no index to can 1 it's about the and where the the boats we used affected is dynamic and Baxter prove that the only building the next to of those 3 types and then you also remain yes and then you are seizing on an automatic trains resilience index on improving that those are the only 3 types of buildings and on that's also you the transfer so you need to say that the differentials well-defined so we also prove that all index 1 index to Saunders is what decisions are on Canal transversely and but you the soul you don't need a family convex whenever you have a problem with the new compact defied modular space for the next 2 cylinders like that and have way more things and not to be able to prove that the square 0 say the home earlier this year you can only do that if you're dynamically connects the region but Interfax said you yeah I didn't want put the zeros here and here could the ball in the front row would ask where those cursor genius and I was counting soldiers yet if he got a little British overseas yes 1 of the "quotation mark he is a trivial trivial Saunders the unhappy do not have killed as actually like holding a vote that it's sort a slightly scared of look at the back of the head of the department of the for a year and as does the move on banana covariant part of the talk but to people of any other questions about what goes into woman corollary a concessionary agreement of theory but because so this close to were stuck on invariants so we need to do something so I tried to define sold the finest of Managua the united with various versions you thought you were think so to clarify this normal approach with bad because you can't prove sort of this sort of 1 of the economical border zones you also don't have on the sort of automatic telling you have some automatic transfers illegal border and was not good enough and you have to be able to prove that the chain homotopy on equation so narrowly go so well so we can do as we can define a non-Apple variant version and when we 7 agrarian version that's kind of analogous in Hambletonian Florida it's too when you pick a arm time-dependent Hambletonian sold all of your home curves and that means injective Serena something similar in the continent world distilled interviews some new things we have to work out but the their and I knowledge turns out that there is a way to take an ass 1 that version of this medical agrarian pictures to prove that its isomorphic the classical cylindrical contact analogy In a very roundabout way get variance for the classical cylindrical contact managers and so on this snobbery there look like on the positive part like technology where is this cylindrical theory if it was the fines on it would look like the the last 1 aquarium version of a positive person .period technology on but we don't want to use syntactic analogy because we'd like to define something directly in terms of the rate data in terms of asymptotic we on suit cylindrical civil morph occurs as opposed to looking at the on poor equations bogus story take James here instead of using 1 sorrow for all time than it takes domain-dependent so did efforts to unionize 1 is going to be an almost complex structure of and we still want for each time he that it goes to a vector field handling 1 for each time see that it's to the tower when the computer and
then we're be counting on while reluctantly Clay counting on still occurs raises throw this the dependence and on because that's not going to know automatically diversity on Manitoba inversion ring had yielded more careful in their arms but the moduli space ran start to want to look at employed .period constraints on is going to be the sky where were on using the domain dependent on them will most complex structure look at stages but it's senior geomorphic equation going on this date depends on please there is my are coronation even as some from 4 nations in the world makes a lot more sense look at the history of the world event the made I was almost as foreseen in some sense arm you can make your land the time-dependent this is like the closest you can get for periodic were busy time dependent you have at its contact like how would you make a time time-dependent this is the context for what this is what I would rather take a with the Union of Soviet where that's the other thing sorry I mean the point is now all the curves there were mostly covered going to move we covered for a generic almost complex searches of the homes of the 2 main sold his news for a generic family and Jay Kim was gamma minus is a manifold on dimension people of the companies in the index of the plus minus the company's interim index of gamma minus and I broke some symmetry so it's got an extra dimension and so you're going to be working in a situation where the so there's a favor by flirt friends of that gives you Trans in the Hambletonian flirt case so this is sort of the you can figure time event Tony and you can pick a time dependent almost complex structure in your Javal Moffat curve equations on and so that's how the curves on any were injected the wagon that India will define the differential of this naive way here so we can't tell you how to define the differential so to find the financial world Bruington needs at all throw in some .period constraints on and this is kind of power forcing a parameterization are rare words in terms of time so that maker and going to be able to look at new prior prowess on 0 and this is going to pick out some wine and my sauna came and you can use this define evaluation maps From our a space instead the image of game was minus all of which redefined To be the women as as to oppose a minus infinity of the projection of the curve on when we were dead time 0 so this is supposed to be like when he will 0 in this at some .period that's in my area were original plus of I and you can you know do the usual thing where you can map on this evaluation met the sensory evaluation that few caution by Oct because even if we do decide differential reception at the model by are for this are is coming from the translation and somewhat position direction from In the end of it and then my chain complex no longer have to throw out all these battery were events but I am going to have to have placed the number of really were listening to the floor so you can kind of think of this as a Morris about version from where he worked from Morris by mail folder rear were written you put a high-functioning honest 1 and so that gives you 2 critical points in 1 critical point has index 0 1 has index 1 source is going kind of that's not really were doing but it's the vestigial motivation for defining a non-active arranging complex so be it generated sinister generators associated 3 so grammar checking down hats gambling has any reward and I'm not just arbitrarily on shift degrading up on my damn enhanced by 1 so we have that the gamma on the image of gamma from considered them counties in the index of gamma Jackie the gamma companies in the index Gandhi had equaled To the grain reversed
so I should have written inequalities yearly but the gradient and Yemen doing universities with heads checks of it so the grading of gamma had also has been able to the Alexander at the grading of gamma check has been able to colleagues in the annex Verena had is just going to be there ,comma Azinger and much of that up by 1 and colleagues in the index of gamma hand you a check doesn't make sense because it's the same underlying reworded gamma should be referred to as just think that I have damaged Czech Indiana had a formal variables associated to reworking of the yeah this is all an eerie it's you also the battery work it's just like instant like technology will also see the variable in this is kind of cool this autumn this is not a good variance I'm not featured sort of not original of but it you know was devastation from survivors while Ashford a Coleman also there's 1 1 2 where and From my differential is about black and composition on and on and on and on and have sort of stuff that goes from check Rewerts Etchegaray so there's 4 different ways through you can go from Limoges right we can talk so sir had the disease generated by this guy the ship so I don't uhm time and I promise that you'll see this evaluation as industry without work half like the claim that you know what that means is that the irony of work also I mean so it is desirable it turns out that this as 1 of version like once we figure out what that's like very version of the sniper very interiors that is attempted a fine and so it's going to have more information coming from battery it's because it's the final resume the entrance of a few chances 3 with you you get the usual cylindrical contact mounted theory so it's a quirk when I guess it's good because you have more information the it's all would do it in these terms cannot so deem check of a check their check to Europe with 1 of the 3 between the degree and index between telephone bidders he won on so this will count cylinders where are you we're going to rates you wanted during effects of points on on the underlying simple reworded associates who also understand some .period an images and also bar is a simple reverberated also and then we want so this is going to be curves you can this model a space going from at Sabina Park are winners the 2 that the positive evaluation map of your curves on the Net the chief also bar and were also going to have to come on the about buildings analysts a cyclic ordering conditions so I I the evaluation map constraints is supposed to be the sort of point where she also bar in these 2 matches with long the highway review of US 0 and done on count these guys and they also have some buildings that we have to count on we've got a cyclic watering condition meaning that some new ones as well you want of minors Of the top curve have plus of the bottom curve and arm whatever point you picking on this intermediate curved closed again on on some the awarded this way as you go around again Serena needs audience here with direction for how you're going around a reworking of and then D check of Elva checkmated check his any something analogous except on instead of having them up as evaluation constrain have a negative evaluation constraint on and on the constrained the it was also checked the Beta had this says no constraint so this looks like the too bad map that you see in the virtually Ashbery I formulation
Soviet get the pluses of Alpha check sorry the head of the church .period 2 points part of bad and 0 for 12 good and the of the check of I had this has constraints at the top and bottom and all of this mannequin theory has not released so easy to count because not only are a sort of counting Saunders that either have 1 .period constraint on 2 .period constraints or 0 . constraints you also need to count more spot buildings that have the cyclic inundation and these morsel of buildings the agony of infinite length because you only have a finite number of a orbits and each time we think the index expert in all they are going to be some sequence of buildings you have to figure out how to count but it turns out that some I'm so if it's a it's the on his that some all initiatives are a farm this the miners here so this also a musician 2 counting on you in an alcove baited with plus and minus constraints and toss buildings it also will count ,comma wastes igloo from an index 3 curved using obstruction Donald ruling that was like a branch of it's 0 you want 1 of the no injuries sold here too villagers to make that work she was vulnerable or is it the soldiers responsible yet this is the way you want political yeah index calculation plenty of the title of the year so major firms are always in the summer objective you don't have the isotropic anymore to deal so for Ply 3 grand dynamic that's for I too and minus 1 the United States this is not an agreement on kinds etymology June generic so is a chain complex In and the analogy but some of it is independent this a family images of your choice of family objectives on the 1 hand the contracts went let me tell you how to relate to vectors fungal kind technology will make everyone would have your
mouth of so if there is sufficient on automatic transfer then and also as needed to define on the classical Central Command Technology differential than we can all thank you for some generic choice and J you can use this today too look at the Managua variant candidate apology could end what's nice Is that all it turned out to be so in this case are DE Chuck looks like the standard cylindrical context mythology differential on 180 kids that all the flexible kind technology differential new and of having opposite orientation but that's not so important OMB but it turns out that some you can show that so if wine J if Wyoming and placement generates dynamically invites J. generic than for the 0 equal to him the queue scientists all I did not use the notice the 1st differential was especially the 2nd differential I told you so dearly plus the minus I and do you want equal to 0 0 multiplicity of honor Donald 0 explained that is the fact you can actually get a chain complex by taking the
canonical variant chain complex 10 Serena with a formal power series on the a variable to that has agreed to so here you in a formal variable and degree With the idea of being equal to that the 0 D 1 this is a chain complex and there is no point in the the people .period it will solve all 4 of the world so many times with hope so this was kind of a wide so you still will be able to see your better orbits and you're not going to have the problems that you had before with orienting them because of you have .period constraints running around the delighted to deal with orientations so I'm doing a better job of explaining the screws or a lot of details on you will be so with the use of 0 . com holders this shouldn't be I should have just written this isn't technically this differential should be considering something like God this is counting mutations really awful the ad will disease had indeed check in there but they have a more familiar expression in terms of multiple cities reworded multiple season here on a single home off occurred so relation of written here is that it is included if we can replace J with a single day then the Czech towns of a you In and J also made a list of Oregon have here on checks and hats with
people .period constraint on the top and you'll get that GE check of all check is going to be the sun over the multiplicity of divided by the multiplicity of Europe could that connects them and my finances become my 1st Cavic but I data With the orientation so the chain complex is still this thing there were using and then the differentials just have more familiar expressions in terms of the previous too fragile for those very unclear and hopeful we can explain the difference in the desert and had been scheduled this saying differences with the board was the head of the was here and you can show that when they do encounter very worried something horrible happens some of the year but I should finish the statement of own and the homology z on the skin complex arm is independent of winter and today in the proposition that's of ICH the tented with you is isomorphic to and so society and Guinean variants of clinical context technology and
and the change in the world and all of I mean if you don't have something as dynamically convex and and had to be working with wood of the or support technology yeah so if you take the connects some of 2 too-tight spheres you dynamically convex so that this sort of result as a work for connects songs but an influential had technology was never built to mean and variance of things that weren't dynamically convection dimension 3 and so on there was a result of the Turbo mentioned his talks where he said that if you have this assumption about coming Zander indices and the positive part and practical knowledge isn't very into the context structure is exactly as dynamical convexity condition and has an analog environment what do you think it His here but you can also computed for some weeks 1 spaces and on way frequent decisions faces the so well I guess indecision words for the 3 stare you have on Q and every even on dimension that's critical to 2 and up in the lens spaces you also may have on the overall 1 plus 1 and you have and copies of human dimension 0 and enforce 1 copies of Quran every even to mention the fate of the the the the of life it was the only about the race so that's what you're doing but you've kind of been able to Cheadle bits of normally your differential would have like an infinite number of terms and on but because we could cheat and use our use of time independent almost complex structure and still be able to define the complex that's why revealed the drops electric terms but that's pretty much exactly what's going on it was the 1st to be embarrassed to admit that someone here to do that and then you get the best of our ability to really really really small press the precise control of the universe has replaced its role is 1 of the more violent place of Jesus introduced the agency that is to say that any maybe but I it's my understanding that you need do connect some you always spend of introducing a new you would realize that there is little we can do about it it led to add to that of the of on no because the my answer is no so I mean it's conceivable that we could define some sort of winterized tents etymology if we were there is another type of building that we would have to consider when have to build understand how the polluter exclude curves many units and you have to take some sort of like augmentation sold hundreds of letters to him so was that he was but that's a garrison you might be able to clear without looking what the foreign anything is possible but they think common strategic platform you get coffee I but at the same hospital thank


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