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On properties of filling of contact manifolds
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Erkannte Entitäten
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00:02
a bit want to and from work on and on and on and on and on and on and when it's a participatory democracy kind of folk so everyone is welcome which was his own fight and the only thing is that this is joint work with Alex launcher that we like to thank the organiser for the invitation the promise and then so they started we sold that states sources in the first one is the paper by yeah Eliashberg and the which is called the towards the definition of symplectic boundaries and which has the following question well if you're given I will be more specific on the assumption that a symplectic manifold so just for a 2nd mass Becky manifold will be open but such that it has the boundary and the question is does w omega determine sigma that so that's 1 and actually the 2nd 1 is the paper by uh when result let's say by the user and Eliashberg yeah sure about this and the following says that if you have and all major which is symplectically aspherical yes really you from remember the other the viewer determined as with any so it's uh if animal may go about things that also know known as well if it's a convex the physical but with the company structure for example with what can you what can you say about that we go make who is 0 so symplectic yes Vericom and the he's able may using that dimorphic 2 are 2 and with the standard symplectic form then and is the from the 2 what would be the ones that we need to know that that this that infinity yes I said that I didn't write it down but they said it had in mind was made that the but that is it's participatory democracy talk so you're welcome to what your favorite assumptions we have or maybe they at the necessary assumptions so that it has and also to modify the conclusion from then it is the sum of the 2 our 2 and actually for any platform for n equal to 0 sorry this is stronger and all the result by Gromov which implies that is actually a symplectic morphing to our fault with the standard symplectic form ahead but this is a very specific tool for dimension situation and I will not talk about that even know what I'm going to explain it could be maybe adapted in dimension 4 to get stronger results in these kind of spirit and so just 1 word about this was 1st it that's it uses the moduli spaces of curves which is the excuse for discussing this kind of problems here and the 2nd is that the proof is actually in 2 parts 1 1st you prove that the homology of them 0 and then accepted mention 0 uh and then you prove that the by 1 is actually 0 which is uh you find correct which was would yeah who uh so we begin here not be interested in now by 1 questions even though it could be interesting to to check that so what we use what you should remember about this theorem is that if it will make a slight art when we just end effector at infinity then the harmonic mology of and is like the homology of afterward and this
05:48
is like a slightly different way to rephrase that which is the following is that if w will make out and that we begin to explain it to be more about that has contact boundary which is the sphere with the standard convex structure then how them is the following figures to what went on to that sorry for the father and it's over you and your it say stays the same of course but to the unit that and makes it more the meaning that we still have again we don't make over pi where does because if you have such a manifold you add this simply isation of the sphere and then you get something which is uh have you something which moms this sphere here inside and then you just have this improvization and you get some money for the which is simply the more the 2 are with the Sun the structure at infinity and into the so uh so this is the kind of question I would be interested in that in fact so let me just say unfortunately we have to write down some definition which are pretty much standard if you have seen much excited complex manifold and if you have an embedding of a symplectic manifold then this is a complex embedding if there exist some form modified near such that site is determined by our fossil is the cone of file and the AlFaisal may go the when of course near near since that's weights the findings and that will come the embedding is exact for some time it said the restricted account that died artifact extensive tool and of course as a primitive of homemade 3rd so that's the definition 1 definition tools which is basically the same is that the symplectic are feeling of sigma is just a symplectic manifold M omega up and I will add with no components such that the boundary of w we sigma when by slight and use of language or we say that sigma CSI is compact embedded in W a male and a site objectives of language because nobody would only exist on 1 side and pair of sigma not on the other side so it's not
11:07
usually embedding OK so the existence of identifies only on 1 side but I think it's pretty I have a pretty clear what it means at the end of this year you can you can always extend by some small standardization so that there's no real there's no should there and that there are so it's exact the feeling is exact the if then again as before but before that see if it's exact as above and the last thing I need these uh uh can everyone see when I write here you can't but it's OK that's the way families is efficient tree that there and the feeling of seeing my side is with its assigned manifold of the if I have w omegas symplectic manifolds outside is a function of is J theories about monarch and J. things omega psi mice Simon is 1 of 0 and is negative on W so it's w is a sublevel of theories about so as a consequence of being employs about monitor there sorry I missed something here and site g star of each and so would so inside the kernel should so the convex form is given by the kernel of J star of the size so if you have a prism mommy functions opening point having makes less or equal to N. and that time simply because if and only if all critical points of size having index strictly less than and think and this is an idea of the book uh no but uh no so is that on 1 of the blackboards but let's start on this 1 by something which is an elementary market based on the from either the ash with urine uh and so some of you may say very elementary remark maybe so 0 is the following so we've seen mount has a contacts embedding in out when with the standard symplectic form and interior components the so if you have a like a service out when it's has an inside and an outside and the inside is causing now let w with all major symplectic feelings of seem there and let's say such that the 1st and 2nd relative homology of w with respect to sigma vanishes you then the map from H D E 2 W when it's a 1 1st 2 HPLC my use by inclusion is injective and the 2nd remark is that any to such feelings have the same make the numbers and the so the remarks is that the peacemaking number of signal the b but the number of w plus the 2 n minus the minus 1 but the number of novel you know there was a will a little bit surprised to this followed that
18:06
so it's not just this sphere and you will see that it's still fairly mean once I tell you it's totally elementary you can actually pull it into lines by yourself but it's not it's just that this here that has a unique convex feeling In this sense unique in the sense that the topology is that is completely determined by by sigma but if you have a class here and in fact if you have a context embedding of something into what when and it's very easy to construct many more so you know there's a very hard if you haven't if you give yourself a sigma a compact manifold it's very hard to embed it enough to win a priori means and mess and it's already given something embedded in Ottawa but once you have something embedded in out when it's crazy to construct new ones by just doing surgery along and those of the index less than strictly less than and you you have something like this it's say he himself to the sphere and then you can add and the like this just as much as you like provided the index is less than is less than an hour and a half times the more with well you're doing so did you doing surgery and so you're adding as Lagrange I action isotropic because I say that them and so my isotropic and so you might find the sigma so once you have once they once you have a signal for example the unit sphere then you can construct lots of things we quite different apologies so the only thing that is get to that she is the ndimensional the whole knowledge in the process of conditions so that you don't distinguish diseases of the poor service the of sorry it's always a con vex boundary I should have said that from the beginning that or my contact structures are oriented and the uh everything here is considered compatible with orientation so we believe that there will come a small number of people so that we assume that the 1st and last time will you attach a hand you you action isotropic and those and make surgery on this sentence because it's isotropic you can just put it in any way you want we don't have my key by still being made this this is something you can find in a paper by the way and all they provide all that by the 18 like this so that when because it's embedded and if you start with something embedded you either have an isotropic and here and basically the isotropic but on the granted so isotropic dimension i minus 1 at most satisfies principle so you can just I realize that as something embedded not crossing it in this sphere and then you you do the surgery and all around these defined in the context of and you can always do that because you read during the dimension but where you have a point in the history of the of the well you could say that because they all have the same as the number here from to consider the but numbers of it number of the usual with the and this is a good thing say that you have a lot of this is just the same as that of the of the 10 to the 4 moles of more claim foremost accepting the mention that you knew I sorry no not in general it let's mean maybe states yet before it proved that which is an so here how many that holds up to 1 of the uh let me state the corollary you probably don't see as great looking what is there a blackboard on which a constant have not kind problem because there itself so I'd so either 1 OK so you have the case but I put myself in between the board and use so the calorie is the following is that if you seem side has a contact embeddings In the standard of 20 and the w omega please assigned feeling then uh and I need to I forgot why but they need and to be greater than 3 uh beauty of sigma estimated number of sigma is the same as the Betti number of w for P between 0 and n minus 2 and B n minus 1 so n minus 1 that's number of sigma which is the same as that and that the number of sigma by flickering duality is the sum of the and of W and B N minus 1 of W so in particular if w is fine subcritical then the N minus 1 of sigma is the end of minus 1 of W and B and of sigma is the no I don't think you have view the means
25:10
this is the resistance to
25:13
his philosophy is here here you would be on the left here it was to be a near minus 1 and yet at the same because it's twin minus 1 dimensional manifold and you have point 3 joining the of the of the of the soldiers also also gives a plus sign the sum of 2 here is equal to to the inverse the legislature was so about that about the point that I have improved the discriminatory yet at the end of this it's it's even more treated almost more freedom than the this can be more than welcome argue right not supposed to write you of the place of married sedimentary anyway a looking maybe I the my morning exercise to students 1 this and the 3rd is so the proof of the theorem is the following just to get w union or when my disease for you take out when you have to remove the Marriott something with longer removes the you get something with malleability W and the and right my of the torus except that instead of sale from that knowing that we to when you when you let is the same most when or uh or the ball so you get this sequence here at each piece of up to n minus z and goes to each piece of signal and then when there is no need to go bad because this is going to be page the first one of art when and so this is 0 this is also 0 for most values of B and so on you get this is either morphism and so for us to get that these not here is injective and then you get so this has to be injected and then you get that the p of signal is the key of W plus b the of art when minus the and this implies by Alexander duality that may be the PT of sigma is DP of W plus B due to when minus peak minus 1 there's a real disease so I don't mean z in the theorem explaining the I now we have well if you can apply this same thing for w we cuisine you use your feeling and so what you get is that the of sigma is the PLC 1st to and minus p minus 1 of the and as a result of the peak of the in the POW well this is actually true for P between 0 and 1 minus 1 you have to to to deal with the non zero homology of what went but let's you that as an exercise in so in this you know this and that and yes of course uh the fact that the the last sorry the last hour of his art went so divided the union yes sorry so it's is not only by of the stories that I would like to do it sedimentary from
31:32
there it's an adverse that starting from a instrument of the effort and that's what it means is that because when I multiply it again because it's no I think because the the Florent might just prove that all knowledge is 0 and then after that actual put the from I correct proved that the by what the by 1 is 0 so here we have and the paper is actually you in your paper and so on on and so on and so that all right with that with that and that that had to have an so we would say it's elementary modes idiot yet has the off as it is in this day and yet Journal of a common model is based looking at uh that yeah that's the uh and various of yeah exactly so uh this is 1 of the 2 so this and this not a well what you actually need is that the Union that picture here so w will you not when minors should be still a symplectic yes very so this assumption on each 2 implies that actually in the paper we got we have other conditions that implies that you need some condition that that would guarantee that these this manifold is aspherical and it's not enough that w when the the view synthesis factor so let me make the number of remarks and when we might not know how much it fits here but so according to some result by meaning out there the if sigma is compact and satisfies that the 1st tranche asset is 0 and W the signs of critical then we have that the cylindrical convex homology of sigma and is isomorphic to the homology of individual relative sigma tensor will basically we become ology of cpu affinity I think shifted by 2 with that's and it's interesting to compare the kind of result that you that you get so here what you get is that if you know the cylindrical uh contact homology of Sigmund and then you know something I mean from this you can extract the homology of the orbit so you basically you know the homology of W and vice versa provided you have this stance of critically conditions in this kind of statement is different you get you ask for any information about the the the contact structure by saying can be made somewhere in this case it's in our 20 and then from that just the differential will reduce the homology of sigma gives you information about the homology of W but was it can also somehow combine the 2 and say well if for example a scene had the contact embedding in enough and then can't you going to know the relative homology of w and therefore you want to know something and even if you have if you in this situation of the corollary where you have this time as the previous time feeling then you know also that the symplectic their content cylindrical content ontology but if you look at the true statement somehow separately they're they're they're telling you different things 1 starts really from the syndicates from this contact structure that you're supposed to know quite well to to compute these and need information about this and the other 1 you have a sort of much coarser assumption by just saying when there's a contact embedding somewhere and then I know the homology and from of signal from the side of the homology of public so this yeah so so this would be a good thing and maybe ends with of of I don't I don't think so so that did me just consider the stuff and on the Net equations and then the back so another obvious question somehow when the question is obvious and mean the answer in the following take n to be a complex manifold look at the content look at the sphere bundle the conundrum of this this view could undermine the of and with the standards of conduct form and
38:39
then the question is whether or the possible feelings of this of rest star around the 5th is the star and we learn that of course the question is whether at least all logically should they have the same ontology as the so is it need to start and where we have the Proposition again I think and has to be greater than trees that if l which all this it follows from what the Fed has a Lagrange embedding you not when you and fortunately I didn't say exactly grunge thing so you have a number nonempty set of examples w that feeling of as far as the number of such that the relative homology of w we respect to the boundary is 0 then from what follows W has the homology something and you can construct of course many of these if you have a which has an emotion you not to and then I'll times 1 for example has an embedding in there are 2 N plus 2 and the assumption of having an immersion in our twenties just no assumption on the tangent bundle so the tangent bundle the convexified tangent bundle is previous then you have such an impartial uh and so with the want to see OK so of course it's really frustrating used to say that all these comes from 1 a wellknown theorem and to my of the torus and and then using well maybe we can work a little bit more and get some cases where there are no known because the few we should try to multiply by the 2 0 so the problem is that the the boundary of the nietzsche start uh this is this term which is OK but it sometimes also the story which is not OK because it's between comes from the from the feeling is that you had so all of yeah maybe I can but if you look look at the aggregate of the so the idea is well let's try to generalize the theorem the firm and yes theorem and then apply a game might of the authorities and maybe we get to a wider class of of examples so this theorem is the following so assume seem exciting and meets a contact the embedding is a previous time manifold now let w the hemispherical feeling such that the relative age 2 of W 1 sigma vanishes then the map and I hope this is correct from a G L W 2 twoway g of sigma is injective for auditory is you and seem outside these just as minus 1 minus 1 with the status theory you essentially get will you exactly get that went up to this by 1 thing which I must admit were too lazy to investigate we get the from the steerable forces planners 1 with a standard
45:42
structure has a contact embedding in our and so it's a political science and with this is that if you have and the feeling then that map from a G 2 of w where g of sigma is injective but this is 0 for most values so it means that this is 0 for music over here this is the 1st time that I was talking about a subset of them was simply sign means that uh what implies that it's a product of c by the biosphere might back is the product of C and I think it's time and for the the so that they can learn simply assign implies that the symplectic homology so that the this 1 is M as a clear sign the symplectic homology of and for make when the response and constant power mansions uh so let me state a corollary of these the following the reverse some of you see your mom is a rational or will just all knowledge is fear and you can see much side embeds in such critical than any symplectically into as spherical feeling we'd be rational homology so when I have plans to describe a little bits so of the proof but they think that's was overview of the state of the art was then used to evaluate the quality of of the world I have the 14th and applied on this clock the ideas misers 20 past uh putting them and so was the boat was too many things so we just keep half of what they plan to skip uh that the little thing I want to embark on getting on the idea deals the proof and things like that and this is where the ingredients are you they're basically the same as well that's a note for this there so what are we we using this so the 1st thing is that uh is a result by Cheney but which says that substrate this time is the product of something which we denote by year uh and and seen so that's how we started the 2nd point is the result by least math which says means that 1 can close and to a manifold the there and then the idea to is to work in P times this to and I'll give you is essentially
51:14
the same as the the 1 man but requires more work and in the theorem by from might just and fair in the national and looking at rational curves which are in the homology class of S 2 so then so what is would actually we do is storybased so what's the and then consider the as will use the was embedded in enzyme see here so because in that bed and now in the time system was to take it answers to remove the and then you add over here over sigma your money for W and then you somehow come from all think cultural the curves which are so somehow at infinity you have 2 curves in B. times this to you use some animal go further hyperplane section to sort of normalize the the holomorphic formal of my fixing 3 points is so that you don't have to deal with the reparameterisation and then you know I mean you are you have also some of homological argument but basically you might kind of continuation you can feel the space V by this by this holomorphic of that is that exactly is same idea exactly z is the interior of a of a node for once it was not of all right everybody has put in all previous statements that was easy and since you complain they drop date but knowledge is it's useful so it amazing a subset status so in in a separate assign and interior easy so here you to to give a with interior component equal to the left and the work is the 1st thing that has you can know you you have no choice on which seems to meet with the with mean that's that's often different and then this sort of phonological argument in the end slightly different but basically the the ideas presented the uh so this is the the assumption that if you don't have to to that this right there you this is the assumption that the system no no because it is something is find H 2 N minus 1 is 0 and then my Alexander 1 the hypersurface we haven't inside and outside so it follows from the from the hypothesis you could be that so there are 2 more things I wanted to do discuss about the problem of this kind of problem the 1st 1 is all simulated that's a symplectic homology obstructions there use of I mean so far uh except for ONE that this sort of starting point which uses uh rational curves contained in a symplectic manifold let's apart from that it's solar ordinary homology somehow mean there's no there's no further homology does that mean that he had his head heat and GromovWitten invariants but uh there's there's not much of a so what can we say free use more about that and they become ology where there was a statement by quoted before by many young but that we see a few more things so insane pleasurable know what these that w will make is well if you I would like to see if the US frightened if it satisfies that the simplex become ology of W vanishes I sort of symplectically and no sort none of us very consortium in the
56:46
acyclic the I don't know if there's a nest is the manifold you get from n which was probably so you start from the separate time you write it and I'm seeing so in in the original and situation of from from Ashford this separatist is just ought to and so it's easy to write it as a product of something and c so it's going to be c n minus 1 times the and then the C N minus 1 you close it as as a combat symplectic manifold and this is and so in the in the original paper it's closes a torus here sorry as stores as a product of spheres OK but it doesn't matter which of those it up and then you have that same eyes embedded of course it was embedded in and Dynesys so it's embedded impedances tool and then you make surgery over P times this told by removing this z which is the interior and doing w consent and see what you get from there by looking at the time of because but I know that you know that you have for the that want to be to find a new idea was that there was a lot of work on the uh on the basis the level and the thing is that that the results of the Vickrey with the blue problem by you and by G but fewer and focus and the sort of the state the propositions here to nerves so let's see like size bounds as a Signed the major let em all major on the ferocity cake and signals be embedded in such a way that you have a separating letting me with the and sigma we separating in and with inferior z and then the homology of Odyssey pathology of z is the same as the homology although but you can and well if you want to know the ingredients where there's a result by 19 which says that when you were embedded we have an exact embedding in uh through a SEEK cure you again for cyclic A. there is that you use the long exact sequence in symplectic homology and then you use result by people who are so this is the the 2nd multiple and 1 shall which says that sigma outside determines the positive part of w under some assumptions you need that then O'Connor's in the orbits on the boundary of the index of anything less than 3 minus and and you can prove that it satisfies here if sigma bonds subcritical response and sold the last thing I wanted tool say are applications which may be in ways most funds for the 1st and also maybe asks so somewhere here what's he that is the idea that you know if you have a workable orders and so if you have this complex manifolds which is concave up on 1 side and another which
1:03:06
is convex on the other side 1 of them is more complicated than than than the other 2 and you sort of increasing complexity in this way uh and so there sort of obvious questions that come up about complex singularities so for example let's add new complex polynomial in n 0 be an isolated singularity then you can look at the singular hypersurface here you can look at the intersection with the smallest sphere and this is actually when he was going to be it's trust so complex polynomial on C and plus 1 so that these values in S splane minus 1 this has a complex structure by taking the maximum complexes subspaces here uh and there's something which is called a symplectic manifold which is the mean fiber here which choose essentially you get by itself looking at X minus 1 0 0 you just motivates so you have your singularity here you have a sphere here where you would like to look at these feelings here but you don't know actually you get something which is just below here and so w w we just be this this this this thing here is w so w is a symplectic manifolds of bounds sigma and w these away just see spheres and it's a wage of new spheres and you is the Milnor number and so here the idea is that the biggest new the more complicated singularity you have me if you have no singularity actually use with 2 to and so the proposition I would like to states is the following and then they fall on the floor again for eventual lived free sorry so might size called link it although the singularity such that the intersection form on w is nonzero then seem exciting has no embedding so no competent mating of course in a subcritical stuff and the 2nd theorem is that if you're looking at risk or you're manifolds and they will do that the think told there are no for example finding that and the idea is that I mean this is some the results for mu equal 1 it says that if you have a previous singularity then w we essentially is essentially CNN and so you cannot have a larger singularities so something which it with the larger mu with some assumption here that we made of course in the what you would like to say that if you look at the singularity we use regular than new prime and you take the manifolds W corresponding to new 1 to new prime you cannot be made the more complicated 1 into the simplest 1 that's what we like to know but they have absolutely no idea on how to do use it when if you have a similar 1 W 1 with new 1 sigma 2 w 2 with new tools so in this 1 comes with SCSI 1 and this comes with familiar 1 this comes with tool and this 1 we can only get tool and then you would say that something like there's no guns that embedding or exact comfort embedding of a graph of the metal oxide tools in W 1 omega 1 the new 1 is less than mu 2 in the sense sigma 2 is more complicated than you know it's sigma 1 1 of the major uh was Nolde
1:09:34
no the only thing I know is is so we use that for assumption about intersections and it's essential for you having patents said you can convince the readers within might find is more than enough thank you much for your attention Thank
00:00
Umwandlungsenthalpie
Resultante
Gewichtete Summe
Kurve
Homologie
HausdorffDimension
Konvexer Körper
Ruhmasse
Symplektischer Raum
SigmaAlgebra
Gerichteter Graph
RaumZeit
Eins
Unendlichkeit
Randwert
Algebraische Struktur
Symplektische Mannigfaltigkeit
Theorem
Beweistheorie
Mereologie
Vorlesung/Konferenz
Topologische Mannigfaltigkeit
Aggregatzustand
Standardabweichung
05:45
Punkt
Gewicht <Mathematik>
Konvexer Körper
Familie <Mathematik>
Zahlenbereich
Bilinearform
Fastring
SigmaAlgebra
Physikalische Theorie
Topologie
Algebraische Struktur
Kugel
Einheit <Mathematik>
Existenzsatz
Vorlesung/Konferenz
Zusammenhängender Graph
Indexberechnung
Inklusion <Mathematik>
Figurierte Zahl
Topologische Mannigfaltigkeit
Drucksondierung
Lineares Funktional
Topologische Einbettung
Erweiterung
GrothendieckTopologie
Homologie
Relativitätstheorie
sincFunktion
Symplektischer Raum
Unendlichkeit
Objekt <Kategorie>
Arithmetisches Mittel
Randwert
Kritischer Punkt
Übergangswahrscheinlichkeit
Symplektische Mannigfaltigkeit
Innerer Punkt
Standardabweichung
18:06
Orientierung <Mathematik>
Prozess <Physik>
Gewichtete Summe
Punkt
HausdorffDimension
Konvexer Körper
Klasse <Mathematik>
Zahlenbereich
SigmaAlgebra
Eins
Topologie
Algebraische Struktur
Kugel
Vorlesung/Konferenz
Indexberechnung
Gerade
Topologische Einbettung
Eindeutigkeit
Einheitskugel
Arithmetisches Mittel
Randwert
Energiedichte
Geschlossene Mannigfaltigkeit
Chirurgie <Mathematik>
Konditionszahl
Dualitätstheorie
Standardabweichung
25:08
Resultante
Folge <Mathematik>
Punkt
Gewichtete Summe
Homologie
Inverse
tTest
SigmaAlgebra
Vorzeichen <Mathematik>
Beweistheorie
Torus
Theorem
Morphismus
Vorlesung/Konferenz
Dualitätstheorie
Primzahlzwillinge
Topologische Mannigfaltigkeit
31:25
Resultante
Zylinder
Klasse <Mathematik>
Konvexer Körper
Zahlenbereich
Gleichungssystem
Bilinearform
Term
SigmaAlgebra
Physikalische Theorie
Topologie
Algebraische Struktur
Differential
Tensor
Spieltheorie
Theorem
Vorlesung/Konferenz
Affiner Raum
Inhalt <Mathematik>
Topologische Mannigfaltigkeit
Topologische Einbettung
Homologie
Relativitätstheorie
Orbit <Mathematik>
Aussage <Mathematik>
Komplexe Mannigfaltigkeit
Grundrechenart
Teilbarkeit
Randwert
Forcing
Menge
Sortierte Logik
Kotangentialbündel
Konditionszahl
Immersion <Topologie>
Faserbündel
Wärmeleitfähigkeit
Numerisches Modell
Standardabweichung
45:42
Resultante
Topologische Einbettung
Punkt
Mathematik
Homologie
Biprodukt
SigmaAlgebra
Konstante
Teilmenge
Algebraische Struktur
Vorzeichen <Mathematik>
Beweistheorie
Gruppe <Mathematik>
Endogene Variable
Vorlesung/Konferenz
Topologische Mannigfaltigkeit
Leistung <Physik>
Aggregatzustand
51:14
Resultante
Punkt
Kartesische Koordinaten
RaumZeit
Statistische Hypothese
Übergang
Gebundener Zustand
Wellenwiderstand <Strömungsmechanik>
Vorlesung/Konferenz
Tropfen
Analytische Fortsetzung
Auswahlaxiom
Hyperfläche
Parametersystem
Topologische Einbettung
Homologie
Biprodukt
Teilmenge
Arithmetisches Mittel
Randwert
Sortierte Logik
Rechter Winkel
Garbentheorie
Ordnung <Mathematik>
Aggregatzustand
Ortsoperator
Invarianz
Algebraische Kurve
SigmaAlgebra
Homologiegruppe
Knotenmenge
Multiplikation
Kugel
Torus
Endogene Variable
Zusammenhängender Graph
Indexberechnung
Topologische Mannigfaltigkeit
Kurve
Orbit <Mathematik>
Aussage <Mathematik>
Komplexe Mannigfaltigkeit
Physikalisches System
Exakte Sequenz
Unendlichkeit
Simplexverfahren
Chirurgie <Mathematik>
Symplektische Mannigfaltigkeit
Basisvektor
Mereologie
Innerer Punkt
1:03:06
Resultante
Hyperfläche
Topologische Einbettung
Verschlingung
Graph
Extrempunkt
Konvexer Körper
Besprechung/Interview
Aussage <Mathematik>
Zahlenbereich
Bilinearform
SigmaAlgebra
Komplex <Algebra>
Gebundener Zustand
Unterraum
Singularität <Mathematik>
Algebraische Struktur
Polynom
Kugel
Symplektische Mannigfaltigkeit
Sortierte Logik
Theorem
Vorlesung/Konferenz
Urbild <Mathematik>
Topologische Mannigfaltigkeit
Aggregatzustand
Metadaten
Formale Metadaten
Titel  On properties of filling of contact manifolds 
Serientitel  2015 Summer School on Moduli Problems in Symplectic Geometry 
Anzahl der Teile  36 
Autor 
Viterbo, Claude

Mitwirkende 
Oancea, A.

Lizenz 
CCNamensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/16290 
Herausgeber  Institut des Hautes Études Scientifiques (IHÉS) 
Erscheinungsjahr  2015 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 