1/5 Contact homology and virtual fundamental cycles

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1/5 Contact homology and virtual fundamental cycles
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The goal of this course is to give a construction of contact homology in the sense of Eliashberg--Givental--Hofer. I will begin with an introduction to contact geometry, pseudo-holomorphic curves as introduced by Gromov, and some target applications of contact homology. The main focus will then be on extracting enough enumerative information (i.e. "virtual fundamental cycles") out of the relevant moduli spaces of pseudo-holomorphic curves. I will present a general framework for doing this, and then discuss the specific application to contact homology.
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I hope this will be good a and some of the time you move in the the end of it I didn't mind and the on the other hand the underground if you think you can call him they still faces from the start on I'll 1st I'm very happy to be here and have this chance to give this class and the goal of the class is going to be to describe the construction of an invariant culled from technology to varied to contact manifolds my 1st lecture I just wanna give us a brief overview of the the general area of contact apology and synthetic apologies to the home of a curbs on I see the number of experts in the audience I hope this isn't on too boring for you that and the 1st of the given overview and it's a lot with the goal so filets and start with a mail folder other mentioned and a hyper playing field so that is economist who wants a bundle of attentive bundles adjusted at every point he developed the hyper plane please accouterments 1 in the contentious base about what may vary continuously until on equally were almost equally to specify hyper playing field you can specify I'm on vanishing 1 phone call them them and exceeds discuss of the crime of land can we need to have both of these descriptions blew up they have discovered yes great so we say see defines a defoliation or is integral to finally have found the lender restricted exceed 0 so what this means geometrically is that .period theirs on the could imagine 1 some manifold and CCI's just attended spaced about submanifolds so locally there at the KSE defines them the charts for why work on sir you have 1 distinguished the so C is called a contact structure is content structure on and we can think of this as being sentenced the farthest from being affiliation as possible so it's often called maximally non-deliverable only on the ground the restricted exceed Is is a 90 general so the arrest saying that produce Lee said is that when the wage of the land of the United Swan is non-zero so USA for example on Oct from 1 so-called standard on a structure which is the kernel of the C plus but I the theater another example on Oct 3 there's a so-called over twisted contact structure which is given by the kernel of and are easy plants in his 3rd year the
yes that's its polar coordinates in some part to the end of the year and ,comma sigh distiller so it's rather nontrivial fear of Bennett and that in dimension 3 views are not the same there is no diffeomorphisms of 3 taking a stand against the overtures to all now on just a few basic facts about contact structures whether it would hardly be the 1st is that was there and says that on every contact holders locally isomorphic too are to impose 1 with the standard context so content structure on just like defoliation has no local invariants on it's not like a Riemannian metric Riemannian metric has lots of local invariants curvature and things like that all contact structures look locally the same so there are only 2 interesting questions to ask global questions which serve in Iraq to the topology of life basic result is that if you have on the repast of contact structures information on for any of the exists and isotopic might itself such as this 1 the formation of contact structures is induced by the sounds kind strange on the technical polemics sales and what it means so 1 way of interpreting is this is saying that the for a consequence of this is that on the muscle space of contact structures discrete takes the space of all contact structures and why the question is no sorry there was good reason you who you are you probably know a sold so I can't thank you yes on a Compaq manifold in the industry and so the consequences that this is discreet and in some sense of this is a it's become commentary object you can you cannot makes sense to ask the question whether all contact structures on the given Klostermann fold the Derbez serum was the same for contact structures and for fully the think on the at on maybe not I will have more than 1 can be other there's no overtures to contact structure on the unnecessary and the use move to those used to evaluate where is the son of a high actually needed a different kind yes so so grace on his true for contact structures and Felicia this is something which is not fails completely for foliations we just have to serve understand with a statement saying Let me say show why it's not true for 4 defoliation there 1st of all if we look at the Taurus and we can define defoliation by began the slow and looking at just the ones of the on and if you very the slope land that this is a defamation affiliations which definitely is not come from isotopic some among the foliations codes of the dynamics of the sort of return that I'd adding by adding having by land on on the circle officials from the basic questions were interested in our existence of contact structures the classification and syntactic ability and corporatism yeah between contact structures of so called on the finest precisely the said so on these questions of our existence in classification I'm not completely answered on my principle on open close the sale manifold was opened if and only if every component there has none of the boundary response so so let why trendy open so then the following are to the hometown people went back to look at the space while contact structures and then we can match to the space of just all hyper plane distributions on in dimension 3 this is essentially enough to specifying the orientation but even higher dimensions Nielsen the specifies that almost complete structure and again this summer twisted by some orientation business this map is a home topical also always wife was to hold the answer to this question face looks perhaps but just as complicated as this 1 of the answer is that this is this spaces described completely on ,comma topic theoretically this is going on the some Cross money and of the Of could mention 1 subspecies of attentive cooked with some extra and this is just the
space of sections of that From that vibration over the manifold is all of these patients in the community the 2 of you hear it for me and this will operate so so so that so here or putting the topology of uniform converges on Compaq subsets so if you're same connected component of this you not necessarily differ market so these innocent connected component but not not differ markedly will be alright classification of 2 start on the topic listens I thought so rifles to that this this discretion discretion the most interesting and Her mostly focused on enclosed close enough for my son enclosed manifolds there's there's some hope there's a version of the statement although it's we have more involved so earliest preventing alliance and the former birds were the 1st recently 2014 is so if you look at all on contact structures from undisclosed manifold so that closed then you can there's a comical repetition of the set into into the 2 different states called tight and I hope interested and again this is a matter of fact time and but this is a home without the gloves on the set of overtures to the back of the bus some was given some extra marking of the server overtures to each region so on but the point of this is that there some this some classic contact structures which are somehow the contact condition isn't adding anything extra work it doesn't have any extra information to the Justice planes ,comma toppy theoretic information contained already this list other sold over twisted ones completely flexible riches on the other hand are far more mysterious itself this is these questions are mostly interesting for protect context a trip that will turn out to be an easy answers for overtures to blunt the results and I believe an open question whether on every irreducible manifold have taken the structure and examples of Nigeria's through no fault of no-take contact structured by but simple I have backed by 2 3 5 country homology sphere and that some of the it's reverse doesn't have any type of structure so there is a there's also a conjecture by Elliottsburg that every every close 3 manifold has only finally many eye-contact structure on this turns out to be not exactly true as as stated on was almost on the way to correct corrected features statement so that this survey cost classification to go and she ruined Honda isn't which states that on any together and irreducible and through manifold so for instance on any hyperbolic through folder any and had any any free manifold on his Thurston Sumatra's Asian decomposition consists of a single piece of land is not server start the easier to understand pieces on only have only has to find the money that the context on and if you drop the assumption of its for oil and there could be there could be a jury has found examples of torrid 3 serious fulfillment faultless with infinitely many the company structures but on the number of homes because of playing fields you get this way this went to the back of of the head the evasive this is somehow a very brief overview of some basic result encountered geometry and hope hopefully just some some more interesting questions to be asked especially higher higher dimensions about the power of this it was the 1st player in the world this year could be yeah yeah so just yet so this summer some standard contact for right down on the ball and that's over twisted and affiliate has inside in fact you can stay the even even more abstractly just for any contact ripped from the ball whatsoever has the inside if only the sort
of thing so you need to stop will go up on syntactic geometry so if you have he hundredfold X even the mention so appears to form is called on complaint that it's closed and so the natural notion of cobras between on contact manifolds for 1 naturalist Ecowas unless there's some tactical Buddhism which is as follows thanks Omega is a Cobras From what Class C plus 2 . 9 per cent .period this and so the boundary Texas y plus distributed minus and when you restrict the synthetic forms the boundary disputes this is the land of 4 on content farms and my personal on and there's also the and orientation for orientation condition which Equitable right saying somebody from gives orientation like it's it's boundary an orientation a contact structure also gives an contact form gives an orientation to the Wyplosz was so I might as it should be the same unlike different so I wanted to know this this definition is not on not symmetric with respect to what possible 1 that can the chorus like this is often often drop like a snake and section of positive and review covers and from my post Lemire said doesn't necessarily go the other way what is that was the trivial Culbertson from an mouthful to itself it's it's 1 of the simplest form of simplistic so if you take arms so this is this is a the books to help these boundary components you can also attach their investments on both sides of that often at a convenient geometric I'm operation today so you can often sometimes catch why crossed the Nunavut 0 incident that's OK and just more terminology I would say why I see that syllable it only takes a circuitous and from From empty so on the the 2 examples like this the sphere with the standard content structure if we take as to and minus 1 thing ball in the end and then has a contact distribution given by the complex tendencies to this to the real hyper surface I'm in general the have a real hyper surface in CNN has a lovely distribution the contact the complex tendencies the hype a service that could amount to 1 ever playing distribution and that's contact if and only if the surface and fiber surface is strictly see the conduct of equal definition 3 stories and long he the yell so yes this yes when I guess when you talk about Corbett covered isn't usually walk oriented the it's always good to be here with you usually what would be itself on this I don't know what it is the new In the you know I saw 1 and now the ladies on the occurs and this is over very fundamental tool in its Vatican Kentucky contact apology introduced him from office so fix somebody's manifold actually for this 1st sent 1st sentence we don't really need a syntactic structure but I want to use it for more complex structure on now although I consider is maps and a romance service index some which are all worth it in the sense that the differential is complex linear have not from romance Texas has called for more victims can only differentials didn't and so on usually sold Castelli means all my 1st we usually considered Northwest bases of all hold off occurs in a given manifold sofa for instance on the
candidate much less space of stable maps introduced by considered and over which is the set of the Paris consisting of the noble Ramon surface and maps from C to acts such that wants to the whole world we also have to fix the homology class farm and should be she was financed by the morphism group and so on so on fundamental properties this fledgling space is that if today and tamed by syntactic former maker which means that of tomatoes positive on complex lines and then this space of stable maps really any any reasonable watchlist of home curves this context the victims has 1 fundamental properties the space another fundamental properties on on coming from the structure of this general Norfolk equation it the system and the lipstick equations and what this tells you is that for the 1st time this summer generously assumptions much spaces such as this is our manifold other dimension which you can calculate the index theorem the question about the fact that this it's a given these 2 that's the compactness and the fact that this is a manifold or or before but you can extract so-called in annuities and variants of this for example some say it's fundamental class and you can live in in the homology of the monthly space itself you can push this forward to the minority of AT Bar present if this is 0 you just counting the number of number of so the this is a very rich theory of Hallmark Akerson sympathy geometry on but I was current contact acknowledges 1 specific example I felt and I'll let me just focus on this example itself certain the important precursor to contact phonology is the best hope for his work on the kind of freaky .period saying conjecture that so many so is please give us a contact manifold and that's also pick a context for on the incident of content form there is an Associated vector field called a better field heartland to break field home defined by the properties of the Member of the river to fill this 1 and the land treatment infielders 0 so this define some float on manifold and the slope is a by contracting office and that preserves the contract structure so is a conjecture over Wednesday and thank-yous 1979 that Monaco's manifold and every the vector field has a closed and of course when Justice question you can ask many many other questions such as all what is the growth of a number of Ray pockets that have more than 1 focus the anti-Serb dynamical question about this fiscal positive entropy things like this the film they may ask why is that the contract conditions an importance of numbers that the Cypher conjecture that any vector field on on 3 has a closed orbits assists also on by character ,comma examples from Coopersburg and in fact the volume preserving vector fields from on to manifolds With no periodic orbits so this summer subtle so geometry Of the contact structure which is conjecture forces forces the closed but is there somehow very special special fields myself but what hope for his work on the whole would hope for noticed on his work on the 3 d wine-tasting conjecture is that the whole market curbs on can actually be used to construct spirit periodic audits of of practice fields "quotation mark so it's considered a syntactical resigned the this just Wrexham again recall at the top end of the summer contact manifold apartment sorry attach these these columns while cross while class costs certain and 1 minus Of across the behind Morgan look at from pull off occurs
investors In this the university can just look at Sunderland money specify what almost comfort structure so we choose and J on this completed corporatism the towns in the interior 5 I just want to be taken in by maker conditions on but in the end on the course some conditions so no 1 should be discussed translation action and in the end we are invariant DS assesses the dark it we want this to both the rate vector field are 3 and on the restricted list contact structure should be compatible with the land so there's some technical commissions you need so it's let's suppose can hold precursor suspected almost complex character satisfies these conditions and so it can happen so this bill this is not a compact manifold I'm so program of compactness doesn't doesn't apply the fixed penalty class Of the curved unit might be nice impact the monthly space might be compact but so let's see how far can we not compact so these levels sets and far 1st see the comebacks likens example of Western minus 1 and in fact any any contact manifold cross are these levels sets are acidic convex with respect to most combat tractor satisfying these conditions so that means is that the whole market Kirk can't escape to infinity stopped not possible to be tension like this maximum principal what can happen those of Homer Verkerk escaped to minus infinity so you can have some some sequence of Hallmark occurs could this be the service substantial of money finger down on dominated and public awareness of and how you can assign to such on a sequence of home of recursive served out a limit in the sense that you get is health care the correspondence with some curves fans this the cylinder the cylinder and it turns out to be asymptotic terrain or on the negative you can sort of see from the definition of almost complex structure if you have a closed orbit Of the Rebecca vector field and trust that with you get up so the whole or six-cylinder in In this impact is in the the symbolic position in our cross wide and the cylinders of asymptotic that trivial saunter over over the report I sold within the 1993 hope for use this I don't know news disapproved of compact Pakistan like this show that a new generation of forward curves and simple positions allows you to define the riverbed like this 1 to prove the Weinstein conjecture in a number of the cases so pretty .period thinking gesture holds for the mainly over twisted contact structures also in the context rupture on the 3 sphere some Jackson by the rebellious bird there's only 1 type context rupture honesty if they get too out of 1 you only need to and unless 1 more example and 3 from an e-mail folder colleges is not reducible so in these 3 examples you can help produce produces a sequence whole Norfolk curves which which have tested the compact on their 1st date dumber negative end of of white cross so during his whole thing so and in 2000 to 7 billion tons cruised from the ones in conjecture 1st 1st through souls in general and those lawyer he proved this was also using home curves but using the whole worth-occurring variant of numbers known as on the CHR American technology to the Hutchings and helps and in particular so what was this this is a variant of contact manifolds which involves counting embedded on home of curves in Our cross the and so there's there's an isomorphism untrained in that context melody and the multiple Clark mulching several written for homology on and so basically some non-trivial the results which are known for her and for ever good informality firstly PCH is non-zero and this gives us conjecture that I actually this is I'm understanding and will that inverting the history this construe this equivalents after he proved the 1 thing conjecture but somehow this the Weinstein conjectured that the work for this was a server a 1st step in improving the sequence and in full so in higher dimensions it's on the 1 stink dentures completely open its it's known and a number of examples number special cases
but it was the inclusion would stand I'm not sure yes it's itself we have written and then expand holdings foreign policy almost all of images on the list you you worry about including the order of the others and that is the 1 who voted for almost all those guys who is to that they stability the solar system because they the hope that the plants don't just collided to the sun and a thousand years so I think that's maybe that's the motivation for this conjecture so this is guess now time to tell you what what kind technology is so this is a variant of the disease "quotation mark yesterday and called for on the paper ,comma was published around 2000 but I think that the ideas somehow those back almost immediately after profits 1st work on this so let's fix the contact manifold wine and but it's also fixed contact form and Hopi artist right admissible almost company structure on our coastline so what that means is these these 3 technical Commission still to be are invariant should send DS Redfield and that's restriction to the contact distribution should be on compatible with the land now principii is the set of periodic orbits the violence that now what this definition is going to be as were going to make some Chang complex generated by the periodic orbits you yeah yeah so sorts sumo on Monday Janet organized the fighting complex generators are all the period corporate With a certain differential defined in terms of the whole Norfolk curves this is going to turn out home to be invariant of concept structure that will depend on the form or the other the almost comfort structure just the content structure fans for corollary just of the definition of the factor definition makes sense is going to be there if we can calculate this very sensuous nontrivial and we get the Weinstein conjecture Singer periodic orbits contact Montes trivial consistent as much income the definition the following so will get on the free algebra generated by this by the set on like she's got all the elements of some subset so this summer issue about orientations in some knowledge of the eh which and stand the issue about they were between multiple covered so there's a natural on partition Of the 7 worries traditionally can the 2 sets circled good Ray Warburton about and this chain complexes just generated by the the good reports yeah Holmes ,comma to make years this is the free on 3 Super communicative algebra generated by so eh vis-a-vis these rabbits have additional discrete invariant 32 either even or odd and so this algebra previous PGA tournaments with you committee on Parliament and the committee so now so far we haven't used home of occurs home Africa's going to define the differential of this complex itself off 4 Denmark held last in I the good I needed tell you what the differential of this city's differential going to depend on the choice of almost company structure so I think the sum over on the overall climate subsets of injury worries we're looking at a number of elements response was faced with a solo homer curves in our across where the top and his ass entirely too plus and the negative and so still minus and the index Of the curve is what plans yes index found so in this case at the indexes fertile dimension plus 1 because of growing money empire you have to use Indiana junior year 0 2 yesterday in this paper because I have forgiven defined the more general theory whether the trouble all home Africa's In here on this this summer's year on so this is an algebra I told you what the differential of the generator is here extended to satisfy all but the role of this becomes a differential created so now come to this definition as is presented by Spanish forgiven on what I want to talk about political cost of the tell you have defined these my joy that counts but the numbers in this definition and so on
appearance that under certain assumptions this on the modular spaces is on or before the 6 dimensions come basically those those assumptions almost it's very often fail in In this example nevertheless someone still expects that these Montoya spaces the carrier for
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solitaire ,comma companies much less basis so this is all but want economy archives the number 1 of the result is that such as as income technologies such as much accounts exists making this satisfy these critical 0 on it so so that when you take the homology :colon contact quantified exceed is and variants of whites is it doesn't depend on the choice of contact former the trestle company structure yeah I think that in also up why is this called homology and was the homology of some chain complex but where this chain comforts come from so here's 1 1 motivation for a definition that so we can look at this place of all that's up from S 1 Hawaii will often just called up the loose space on and vector-one quotient by politicians of sorts of loops and wise up to the parameterization wrapped up rotation as is a matter on the given by sending even in the gamma the integral part of the contract for so many of the if you have a contract form you can look at this distance integral itself is a complex quality is very very roughly speaking version of Moss homology on this space With this on with respect to the structure of the the people who whole loses his yes yes votes that somehow the space of states of the strain in the yeah yes so if you and some straightforward calculation shows that the critical points and dysfunctional are the requirements periodic orbits and the gradient formally speaking is exactly the see the whole off Kirkuk question so this is going right it if we if we just have to serve the free to vector space generated by the periodic orbits this would come comic module will exactly be Moss floor homology for this function so In some served 1 can study more scary on the loop space and serve much more simple examples like this than than this if you take take a the people at this level of the decision doesn't matter much that was 1 of the following the yell yes was just as delighted that so serve more common form of doing worse the analyst spaces to just go on and get out of the Uvalde and like more scary this is sort of the testicle the technique of selling those spaces to on studies served Moss the analogy public space with this functional so that this say why this is why these 2 somehow very different so that the critical points of this functional Our close duty 6 so maybe and maybe a lot of money protection here on critical points of these are close to the 6 but more is to be sold at the we have a functional critical point you you look at the Hashim so at a given close to the secure does the Hessian some salsa .period operator has infinitely many positive Eigen values and we finally many native I can value and some of
the very very much related to this fact is that the downward gradient of of this continent over the years the energy ,comma is a parabolic PT it's basically equation an unusual new version of it and what this means is that some the the initial the initial value in particular the initial value problem as well post fetus because any sort of random loop there's a well-defined downward for granted radiant floor of this energy on Starting about good but this is not the case for this fall here so the downward gradient all of this function which is usually a the action functional this and has a whole month equation which is elliptical and as a result there's no well defined downward philosophies are just a random will appear this notice well-defined down and so what what what this means is that so the consequence is that on but when you do this isn't for isn't appearing inspectors it's it's not all that doesn't calculate but the homology of this expects so if you are just trying to stay that the loose basis the servers and Moss analogy works and you can get existence their arms stroke close to the 6 on manifolds by 5 I'm saying something about the topology of space and using this version of more more scary this is actually gives you some sort of cell the composition of of the but this this doesn't on In fact ,comma it depends on the context structure you choose them otherwise it'll be served warning vary but that what we will be you all on the the institutions from civil war that most of the alarm I've been in that I think it I think it should be but ,comma fathers would be there haven't been any local mobile home yeah yeah still it's it's been done like there's a there's a whole lot of stable home because of yeah there should be some something some of stable stable home happy something you can get counting the Maasai space of higher virtual dimensions of not just wanted to mention 0 yes there's another reason this works is that to do to do more scary you don't need the downward gradient flow to be well-defined well posts all you need is the the problem of surf specifying the critical point to the top and specify critical point in the bottom of this space of flows To give you all the financial aspects and this this is well-defined free elliptic PT cyclists so even down flows and well-defined we can still serve to 1st of March 3 so right if he tried to serve take this definition on and to do exactly this kind going to get something called cylindrical contact mortgaging his itself and in some cases it can do is take on the complex which is just a the 3 vector space generated by 2 5 and take a differential which just counts against the spokesman that the the discount for example and exploded in American life I hope you will lose it is the duty of all of his life could be it could be with a wee wee due to counter requirements also here what we the farm or yes yes so this is somehow it is yeah it's this is this is a different a different theory which exists unless generality Friday on Anglesey what this is this doesn't work the definition and in general get results so this is somehow the most direct idea of taking Moss burst around the loop space and turning it into a homology theory why does this work also the homology theory you need for this square the differential to be 0 so in all these types of Moss for theories 1 1 when the Socialists and the reason is true is the following so on it if you look at the square of the differential applied to what plus
Was this was just counts of two-story Kurds you want plus going to some any here the and then ending 1 minus can raise both the index was so why it why should this be 0 well the favorable only 1 said is that this is the boundary of the modular space of trajectories just starting the pluses and ending at the time next so this much less space and is generically Derek one-dimensional and it gives us the call risen from this modular space to the FTC and particularly its follows of this this coefficient of 0 the show another problem is this is a list the boundary was space has on the outskirts like this but doesn't only have curves like this it also there are people also have a splitting which looks like this and so on you no longer get these critical 0 you could use chronicles something up so you can you can define some sort of a series of operations here on this vector space from the 0 Gamma counts .period bases where you have no later than the 1 can't say much less basis where you have 1 negative and the 2 kinds of things we have to marathons etc. so this says here some of the 1 squared is it the 1 squared plus the 2 0 0 3 2 equals 0 so if you're if you can if you assume say there are no contractual Rabe orbits that this degeneration can happen and cylindrical come technology is well-defined but in general we have to serve count 1 positive and an arbitrary numbers of negative ads together well-defined penalty-free and he the recent and read the reason we can't restricted as 1 positive and is because of his maximum principal ,comma I've talked about earlier so if you have on the show ,comma is Simitis curves can't escape pods the positive and so if you start with 1 positive and you can never degenerate have more positive ads we always at major events also have screwed to resolve it's embedded comfort Maltese is defined in dimension 3 and dimension 3 of our crosswise dimension for and so and the condition of being embedded the FBI will serve well defined in this section vary between home of curves in 4 dimensions and you can people get embedded Kirchner gives you very the solo restrictions on the sale of curves you consider and this gives you a different from all the theory which is isomorphic December separately 3 unfortunately doesn't mean it's not really clear how the generalized CCH to the higher dimensions and there's no corresponding nontrivial results for contact knowledge so it's simple applications in this history so pressed before I said earlier this is it this is not then Wednesday conjectured holds for whites the owner of the see the things that distinguishing different context structures the is another enormous certifying status quo and so is fact that the contact multifamily over twisted the manifold 0 it's license to join particularly concocted from technology for something and is not 0 than the start of a trusted so I system so begins in any Florida system manifold so you can find them I can't find a for and Jay Kester the history sounding exactly 1 always playing the innocent position was like this section so that the construction of the services on that
LEI Ashford dimension 3 of were shot in the Kruger in mention on higher dimensions on what this tells you is that the differential of this guy 0 2 1 is a boundary in this chain complex 0 1 equals 0 in and commodity To some all you this 1 will plane which exists locally and the standard Over twisted all comments and contact knowledge of anything over vanishes so there is actually a common thing that happens this reserves should be expected to in sympathy the contact geometry there's some classes of structures of some classic problems on which are flexible and over twisted context riches is 1 example of that the 2 classified completely by homotopy-theoretic data and 4 of these sorts of flux problems hold what occurs generally tell you tell you nothing more than the whole effort in the car ,comma variants are usually trivial and ,comma originally results and in contracts but it's apology often proved using you may hold off occurs and finding instructions using curbs so as as a result a combined these 2 results concludes the Weinstein conjecture holds for anything overtures so what is actually to go over 93 for 4 3 manifolds and property over 2009 Weinstein conjecture calls for overtures to you didn't think Yesawich history so know thing even tho this is a constructing Phil ability so the has a contact simpatico worsened from what plus 1 minus you can come home off occurs in the corridors of the with 1 positive and the rebel revive plus and the genus 0 on bonds of negative and they were 2 1 minus and you get "quotation mark Colbert is about going from technology from 1 class to I'm innocent homes and you know something interesting happens if if I plus is over it is owes over twisted because then what possible to stand up and say so that if a positive and has 0 come technology then so is the idea that is that this is a unit ring rapper has send 1 closer In particular if if you're over twisted you're not available because that would be a Corazon from your richest amenable to the empty set in a kind of multivendor said yes it is an interesting question whether come technology detects overtures to the since it would be it's amazing if there's income technology on implied that you're over overtures but the fact that he used to say that Huge for additional reasons to give him the my master of this Yukon commuter who will also have be to to with this is what they want to go to they can definitely but I overtures to can definitely be capped In his arms yeah I mean I mean assuming the cap exists like topologically the parents of the involved 4 the overtures to attend cat final no I think I write out Alaska Murphy have this prepared like constructing ,comma like making making cobordism syntactic Ray I think this shows and the shows the negative and if you have if you have a cold waters on like this and you have sort of the contact structure here why minus a C-minus and if you have the a formal some like on almost emphatic structure here almost context Archer here then you can make everything syntactic in contact as long as that the top boundaries none at the end of native founders over so then you can the new industries the possible ambiguities fearing capital again we would all like you did yeah yeah definitely added that only this is something you want to change the orientation of the Society of London you have to change the size of the French plans change almost completely cured so he saying all of at the
same time it would be there's also this is the interesting questions and the corollary of this would be that if the if the positive and of competition is over twisted dominated offenders is overtures to the masses the widely open right now it's known in dimension 3 if this is a Winston incompetence some of very recent turmoil and the 1 that was found in surgery and mentioned 3 preserves tightness on it would follow if we knew this but this seems this this organism much harder harder statement apparently that completely open higher dimensions so there's there actually own there's a related invariant on this side if you have was underwritten in part once there's a relayed invariance lasagna and content technology and there there's a related motion foundering being loose and of over twisted and dimensions at least 5 cent and 4 of release was under incivility that was under incomes acknowledges 0 just just just like this so you can ask the same question and and that's also told the recent examples of to be 2nd found on the Sundance and hard to impose 1 at least 5 of which are not loose but that those under contract technology is there so this is this is probably probably false and higher dimensions no examples little by little we give some examples of calculations of the fund technology so the the easiest case is 1 you can find a contact form were already and have even greater because them this unchained complex is concentrating even agree the differentials degree once has developed so this is new actually this and some some cases so that the Lasky found authorities even contact forms uncertain the from fears fiercer links links of on isolated hyper resurfaced singularities and then the corollary of this is that they're infinitely many play context ships and ask for it was come because have become due to different of course this also implies the Weinstein conjecture for these contact structures he can come entire certain growth rate of the number of reported for these contact structures Harry everything now all the server applications the something here the on compromise Carini showed confound even content structures on 1 just across the street and they can also distinguished by using content from all the and and and many others so there are many other calculations of come technology a lot of that happening there before the you rigorous definition of mulching I it was in place so that I believe most of them should should work with this definition I'm good at give but found I mean or whether there lots of them I have arrived on those of us that they think so I think I mean 9 of like 99 presented of the woods what's done should be this completely valid and what what the result Allstate work made more precise version of a result in a 2nd but it says in particular that it's all the whole curves are cut transversely than the cup counts are believing their review of of the contraction in the summer the and the new however let me just mention 1 more applications on synthetic embedding capacities so if you have In some open synthetic manifold saying some subsets make old only 2 1 animated to insiders are 2 and the standard syntactic structure interesting question to ask is Is there are some but the embedding of 1 into the other law and this time also known squeezing them says that if you can bet on it and that's a ball of radius thank the then all are consistent they got so this is the 1st evidence that this is an interesting question and you can you can
find obstruction still alive is not obvious obstruction the volume this has finite volume for any final and as in the volume but yet there's some non-trivial on constricted if these contacts amounted together nontrivial constraints on the following way and in fact the height a lot of already embedded their capacities and which come from Bennett high-technology park are known and had then used a lot of I think nobody's really studied kind technological capacities that much of the north African technology capacities so if you have an embedding player may go on until they get to you get service covers all maker to -minus Omega 1 it is a covetous From now make to the president and commander of major 1 and so on ,comma technology the recorders amounts from contact knowledge of the boundary the cost of maintaining that it is going on yes I guess who yet so it's yes good so it was Sunnis have contacted boundaries and so each of each of these groups has a filtration and by the action that's this this Moss function that not that this more functional were from Sheridan earlier the integral of context form on and this map has to come 1st to decrease action by some stocks their argument and this other hand if the strikes convex domains and then you know the kind technology of the 3 sphere and on you know this map has the ice morphism so on 1 hand as nice morphism on the other hand it's it decreases action so if you can somehow calculate the ex infiltration on contact analogy Of of these the boundaries of these domains gives you obstructions to 10 1 of the other so hunting season saying that sovereignty and elements of the Contract small Tuesday as 3 you get something called the capacity which say sees the wife of it later which is just the minimum actions such that this element time is represented by some rebarbative action lesson what was that and this and this number is monotonic with respect to synthetic embedding I'm so that's all the applications of me just to conclude by work more precise version of the theory that all the spends some time proving the rest of the class so before any Wilander context of the contact form and an almost complete structure closed manifold on the Justice said data from just on trail in the input data along with 4 every element of assets From the Collection of numbers for repair which we just think of as the criminality was much less space such that when you form the contact Malta differential using using these muscle accounts together this critical 0 home studied an extra condition I tractor conditions because as stated that there was vacuous so 1st on December the none of the it's still not because I could take all the muscle accounts to be 0 was satisfied these critical 0 so so that he other conditions of them imposes just said if this much less space is on transit benefits cut transversely then the ritual counts are important shortcuts that you expect the cut transversely is this technical assumption you need to show that it's a manifold of of expected dimension yeah for each breach of data breaches of profit so when you have a bunch of 9 transfers yes if you have you have a single man transfers much less based on its virtual fundamental class homology is well-defined but if you have a bunch of my joy spaces and they have boundary and the public the boundary of the given much less space is a product of a bunch of other much less bases then you kept kept on turf stay with these numbers are economically you have to make some choices perturbation choices in these choices on effect of the answer you get these are the status of data somehow set of all perturbations you could you could make how to make things transfers and analysts is that the resulting numbers if you have something already transverse you'll need a perturbed and even if you did you'd still get the same answer so that's sits on offices so there and their corresponding
theorems for syntactical were sons and for families of some political prisoners and these are necessary to it show that this is this isn't on invariant found button until just status the favorites essentially essentially the same only so so next time on all defined the the much spaces in more detail and will start the proof of this construction of fundamental cycles few and now