Symplectic embeddings of products

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Symplectic embeddings of products
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McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an “infinite staircase” determined by the odd-index Fibonacci numbers. We show that this result still holds in all higher even dimensions when we "stabilize" the embedding problem.
Principal ideal Standard deviation Moment (mathematics) Sampling (statistics) Differentiable manifold Line (geometry) Time domain Event horizon Dimensional analysis Subset Product (business) Many-sorted logic Simplex algorithm Manifold Einbettung <Mathematik> Modulform Right angle
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and and so on I'm going to be talking about inflected bearings of products this is all joint work with Richard Hines let me start by giving you an introduction to the kind of questions that I wanted to talk about so if I'm given to selected manifolds and 1 and 2 men and a simplex taking batting so give you a definition sympathetic amending it is a /slash going from M 1 into and 2 where side is embedding of smooth manifolds and 5 pulls back the simplistic form on the target simplistic form on the domain so 1 can then ask the following question which is if I'm given to some like a careful that like to know if I can find is inflicting embedding of 1 into another In the event and 1 from England and to make it to does there exist a simplistic complaining so here and I the right lines somebody combating just like so yes here just stands for some plastic so this is the basic kind of question that we wanted an answer to it turns out that this is quite subtle and uneven example to illustrate why the little in a moment but anyway so that that sort the basic the basic fact that much of the money at the moment in samples of generally I should say that this is particularly subtle when the the dimensions of M 1 and M 2 and that'll certainly be the case in the stock if the dimensions are off by safe for there each principal inflict we should really think of these as having the same dimensions but when I give an example you'll see so many other questions about any of that police To be pretty straightforward but it's let's go 2 dimensions for now and like I said this problem can be fairly subtle and the following example due to McDuff and slight is particularly illustrative of the the some of you couldn't have seen this before and I'm sure some of you have not and will also be quite relevant when else replying that the top so let me bench might want to know is they I want to know how this impacting betting problem work for some very special shape In particular the defines the four-dimensional practical and so when I think of this as a subset of which identify with our 4 with its standards inflected forms and and I also want to study the four-dimensional ball thank all right of course which is just a little were a and B of this thing and they wanted to know when an ellipsoid admits is employed to combating into a ball it turns out it turns out that we can encode this in the
following functions so we can define a functions which is
given by the In the light of real numbers such that there exists doesn't like to combating from the 1 In 2 a affordable besides Linda OK so this is a function from Arda are and it turns out that see that for a predictable 1 completely answers against called him 1 of the questions that come up later in the year for a great called want answer he won while that might not be immediately apparent if you're not used to thinking about these things before but the point on is this infighting bangs and scale them and they're also symmetric new EAB is the same as the baby so all I know is that function for a greater the 1 that determines exactly what it was like to call so we definitely were able to actually just computers function and Elcho the answer and if you haven't seen before where I think you'll find it pretty cool so well when we scrap the function for you it's like it's function eats a real number and spits out a real number a basic consideration is that the man is always greater than or equal this is where today this represents the fact that complexity embedding have to preserve all year why no sanctions what the function looks like it's all draw this word of a here here's and we only need to know what's going on for a greater evil the 1 so at 1 of function is 1 and function that is I go by out a line for a little while so if you extend this line it would go through the Borges and then at 2 well the line years off to the right OK so let me say a little bit about geometrically what this is saying so this occurs as 2 and this occurs at 4 so from 1 to you should try to visualize what's going on especially if you haven't seen this kind of thing before effort 1 2 2 this is saying that this embedding problem satisfies extremely strong rigidity OK so here I have an ellipsoid maybe haven't even won 1 . 5 and what McDuff slings computation is saying is that the optimal betting into a ball comes from including it into a ball it it saying it from a between 1 and 2 we can't do any better than inclusion the home of OK Augustus was known to but I mean if you're if you're think that is betting problems for the 1st time you should think of this as an expression of quite fun rigidity however aptitude of functions suddenly veers to the right and and this is a horizontal line until I can't go anymore so here you should imagine ellipsoid that's 4 times as along as it is called and this evidently can fully fill a ball of sites to occupied 100 per cent of the volume and so will then of course the function can continue anymore so what is it well it does the same sort of panic it continues by alignment of the extended it went to the origin and and then it veers off to the right it does this again and it does it again and it does it again and again and again and it doesn't enjoy many times and until it terminates at all this number here is the golden mean to the 4th power this is the golden mean to the 4th foul so now what is it due after the Golden into the 4th firewall this is about 6 . 8 and actually it's also extremely interesting after 17 overs 6 square which it should be obvious to all of you want to see him with the gold Nunavut and so this is about all this is exactly in 36 so after 17 6 where the pontiff is the volume so here the serious is just equal to the square root of it I'm interested not in between well it's kind a mixture of this behavior it's mostly the volume but there's a few times where rebels in its obstruct so actually right after touted the there is a rather long dished out but I think there's a more steps they get rather small and other than that it's it's just the volume so so this is actually the peace earning most concerned with in the park and the use of beautiful results so a unit of the music and applause and see if you have any questions that I guess the way you should think about it is that if the stretched enough men all obstructions vanish except for the volume obstruction but the look so it is closed around then there's this very delicate and of course there's something else I wanna say about the new you're going right there is some very delicate behavior so of course is something else I wanted to say that this behavior so late said this number is 1 this numbers too this numbers for we asked What are these numbers here but some take a little thing like this and blow up repair so if I have a stunt like this while the pattern the a call the ax coordinated so here the extraordinary this call and this would be a and plus 1 and while also what the fans and the it turns out there determined by the odd indexed to the notching so they're both ratios of art and at the notching numbers the area and so far where's the ratios of consecutive on an exponential growth in the vehemence look like this you in here the thinking and adjust and these are the things that so because there's this staircase determined by the Fibonacci numbers this is called the Fibonacci stare yeah so apparently the legislators closed around there some delicate ,comma tutorial I'm behavior so but any questions about actually what I think is the minimum thing rarely we are not but for example at the border the you know this is the interest that if you can't find anybody of a closed the 1 for into a closed the 2 you can find in Bellingham and opened the 1 for into an open meeting and that this is saying that you can find an embedding of a closed in 1 4 into the two-plus at the open we open 1 of the things that you could also use right this function you could write this function with open ellipsoid isn't and open balls managers would always obtain the more questions the result of the body actual maps yeah that's a very good question so that there may be a few these you can write down but in general and this uses a technique called some deflected inflation which is certainly rather non-committal on the way and no I don't think you can write down the 1 for which the information of with another you and I guess you can probably do the 1 and squared things what can you do it the square to be commuted exclusively that the other cases like the unit is that of work OK so that's part of the four-dimensional story and while another part of the the story that I wanna say and while the various
ways of thinking about why this is true but and the result which is going be of most relevance to this talk is that result from slightly later also due to do so so a little while later disavowed purely ,comma Troe criterion for explaining when 1 ellipsoid beds into another so USA showed that cinema-going open ellipsoid steered the cleanest state then there's some and 1 and if and only if well this sequence NAB is less than or equal to the sequence OK a minute I'll tell you what these sequences are and but this is just some sequence of real numbers in here less than a equal to means term by term inequalities purely commentary sequence determined and in fact what it is is so NAB is the sequence whose case terms it is the Hey plus 1st element in the matrix and many pluses and b I where them non-English managers and here I mean the capers 1st smallest element but this is indexed starting advocate equals 0 so what this is saying is given positive real number is Amy you take all possible non-negative integer linear combinations and the new list them in order including repetition ,comma 12 sequence and using the common interest expenses and completely determine when 1 four-dimensional excluding so just give an example of how this and works action admitted to much of the commentary said the end sequence the stock by 2 or itself for example if I wanted to end 1 1 for the 1st time would be 0 OK because take 0 times 1 plus 0 times 1 and then get to one's take 1 times 1 1 times once they get 3 2 and and well so so you can associate these ,comma trail sequences and indeed you can see ,comma this resulted from the Fibonacci staircase part by using some of that was coming .period so what is relevant to our story is that this sequence NAB well it looks kind of somewhat beautiful but somewhat render it has an interpretation for us that will be relevant this is the sequence of the CH capacities 1 of the Calypso added you the so he action against called embedded content knowledge capacities I spell it out for you this ended up being Messerly written but the users sometimes called the siege so will return to his ch actually quite but what embedded content Margie isn't the variants of the kind of simplistic field theory ideas that have then discussed here and it only exists in dimensions In dimension for contact 3 manifolds and they can use it to get four-dimensional some to combating instructions called the siege capacities and from that point of view McDuff results says that this obstruction coming from that account technology we shot for embedding looked so you don't understand too much about this this will talk about the siege again relatives the question In I know the annual cost is would be the person asked but my impression is that I am where In light of this OK so this story mentioned for and and I guess I'll briefly remarked that of course ellipsoid the rather simple shapes you can push the they give a little bit further and so for example with terror home untreated press Alessia Mangini we've been seeing how the story would work for embeddings of some deflected toward forming and things like that toward the means to pushed the story somewhat further in it's certainly not the case that we have a good understanding of when 1040 form in folding beds into another but some of this unit does generalized I swear and power what I want to tell you about staying there is a story in higher dimension so In hired mentions the situation is much less poorly understood even for ellipsoid that site more poorly understood even in cell well so for example you could ask When does the ellipsoid NBC what went into the woods you well you could maybe said I should look at the obstruction coming from embedded contact knowledge but I like it is said there is no obvious obstruction coming from and that account entomology because there is no known or the analog video content knowledge in higher dimensions of there's no Kowloon panel the Ch Of course there I'm obvious in a lot of this NAB sequence and you could maybe guests you could still try to guess that this commentary sequence still understand the embedding problem the good think that perhaps ice positions with interiors here you should think that perhaps a result like this should be true you could think that perhaps a result like this to be true but unfortunately this just isn't true so even this is not true so Wiltse see actually a little bit later why this is not true but from some of you may have heard of the work of booth on the non-existence of intermediates and talk to capacities of inaction used to show this for example this isn't even as common interest sequences is not the right thing so I think it's fair to say that even for these ellipsoid were pretty confused about how the embedding story works so With Richard Hinds we've been wanting to understand a more modest call so 1 understand the following more modest goal so we want understand well we'd like to understand the following functions that's an analog of the functions study by the death of intellect 3 and a fine the will To be a wealthy for Rolando such that an ellipsoid processed With an odd to us simplex deeply embedded into fourball besides land crossed with an hour to
an 2 sure we can fix we
understand so if you like this so upset of course the talk is called some plastic embedding product maybe you can see why the talks ,comma now but he like the ellipsoid paradigm you should think of this as some ellipsoid and where most of the factors just said equal to infinity In similarly here so this is just some question about the extremely stretched a little so while now the basic observations is that this function see 12 it's always less than or equal to the function study by McDuff and life with that's because if I haven't embedding of my four-dimensional leading football and just take its product with the identity come and get the embedding of the product so in a way this is a question about how much flexibility 1 has from bodies of products can 1 actually do better than the product OK wall so you might I think know slightly wounding of jobs and again so you might think that but in fact the twiddled is equal to see McDuff flying today and in particular that it would depend on it however that's not true so for example there's the following theorem Due to Richard Hinds earned which is a kind of refinement of goose construction so what's wrong with it so high and has shown that there is employed to compelling From well and he the 1 pay cross are to hand a formal of size three-day a parade was 1 and cross onto so well now I'm actually dropped picture for you it so soon at this point maybe some see why says that this can't be equal to the McDuff slang function but I still want drop picture images can be very important for understanding what's going on so I know that the McDuff slang function it's all created the equal word well create over a plus-one converges on as a ghost to 3 the case again I'll put this in the world form but this says that eventually 1 must be able to do better than what McDuff and slanted and because there is no volume obstruction to this problem for a so by the way if you used to the idea is bigger than what it was and and is bigger than the 1 year so again if you're learning about this for the 1st time you should think of a as you know extremely large maybe even infinity and then it's kind of amazing the fact that such a thing exists because we're squeezing an entire are 2 factors you know we have an extra we have essentially an extra factor of 2 year suspended amazing and I like to answer something that the bank said but this actually can be done explicitly this is constructed explicitly using a kind of catalyzed folding Clinton which is probably meaningless forward maybe even 90 feet per cent of the audience but there's a construction and undue deduce fields like others called some plastic folding under any can't quite you selected folding to prove this but you can use a kind of catalyzed version which is very similar to quite beautiful art especially try to understand what the heck groove is doing lease for some of us as can a lot easier to understand analyze so the very explicit about OK so anyways now want a dry picture yeah In but yeah that's a very good question yes it's at the moment you are only most of our only ways of producing embeddings them come from some explicit in higher dimensions come from some explicit constructively and of course 1 can take a non explicit construction of medieval into a ball and cross the yeah and in dimension for we have these inflected inflation tools which are very powerful and there's not a canned good analog that she has most of our ways of getting canned nontrivial higher-dimensional support in Bennington freehand on yeah so I will get back to the shop question a moment and but the at the moment I'm just saying he constructed 1 and if you think about it and that shows that the McDuff slaying result cannot possibly the answer here so let me drive pictured underscored that this picture is just supposed to emphasize so I'm now going to try to graft new will well the graph the UN 3 a overrated plus 1 why just have start you know that 1 its asymptotic 2 3 OK so here's over plus 1 now square root function where is where
function and in particular it crosses the grass 3 Over the last but some value and remember CME the McDuff flying function is always above this white whereas what we're saying is that behind result says that whatever function the travellers is always less equal to 3 or 4 over a possible so eminently once we get past this point OK this product function cannot be equal to the McDuff flying functions so now natural question is what is the point was kind of fun exercise anyone good mental math what is the point where the Rio was the airline for which they have rapist 1 and spirit of insects well it's nothing other than the golden into the in the so this intersection point is the golden mean to the fore and in fact if you draw the Fibonacci staircase what you'll find is actually bounces between these 2 Graf so if that is the case with something like In other words the upper graph here does clipped the top of all of the steps OK and so based on this kind conjectured in an earlier paper and that seat would always actually equal To the McDuff Sheline function them for a between 1 and 10 the 4th and rich I proved that was the year it's to rich a 9 is that the twiddled a key policy but work I want to tell the again if you think geometrically in terms of what's going on it's kind of cool situation this is saying that the optimal embedding In the Fibonacci staircase region is just given by lifting the McDuff flanking that is just given by product in betting and then that's the only part of the McDuff inch line graphs that persists to this problem once you got moved past the Fibonacci staircase part the rest of the graph is not real some of them are not used to really is the robust feature of the and so I know that's a great question yet so of course this says that the twiddle doesn't depend on and from 1 to 10 the 4th Richard are currently thinking about what happens after 10 of the 4th and we don't know the meaning it's actually a funny question if you like trying to figure out which modulus spaces of homomorphic curves are empty were not and so I would guess that it does not but I have no way of showing that I should say actually there are few more points and that are known due to Richard so actually also if a is equal to 3 and full 3 D plus 1 would use an integer then the then the the graph is also due the held loss the few more points on the line the 3 overrated plus 1 graph lines so you actually do know if this is a equals 3 D plus 1 where D is an integer and greater than or equal to 3 equal to 3 times then you actually do know but these are correct values these also took the early resulted hind really early OK so I want to say a few words about How the proof goes it's very good news for you because this is a conference on Model basis of homework the curves and the proved that involves a lot of tinkering with modular spaces ,comma Africa also interestingly it uses embedded content homology I'm quite a lot which is interesting because of the higher dimensional higher-dimensional problem are there any questions about this so before and give some sketch of the proof goes my I want the economy it's kind especially look at this folding construction you'd maybe never guess that's ever shot because it just looks like you can do this and that sort of amazing that this is somehow pretty persistent feature craft that actually it does seem to be with with terror home and others we've been the about how the story would work for other state toward targets and this kind of part seems to be a pretty persistent feature of what's going on and there's also some very interesting ,comma networks in Tallinn the 4th and involving Erhardt polynomials so you can find triangles with slope within rational slope given by bite out of the 4th that their .period enumeration looks like at the latest .period enumeration for injured triangle that something very special talent the argument that the state would have been going on year to In London said that right and factor is going to say yes it's pretty similar to the high-income an argument but with a new input from you see for holiday so I'm here they sort of do but it's more explicit they do some sort of next stretching OK so but actually I wanna introduces question even write something on the board so what's the idea and well some of you are probably not used to thinking about this employed to combating problems at all and so municipal the son of a little bit of detail so it actually suffices to show that's the twiddle of Libya and is the same as the McDuff slaying functions and the media that might not be immediately apparent but that's not that hard to show so what I'm saying is remember we wanna show that the grass is given by the purple thing purple in honor of Joe and so want show the grass is given by the purple thing well what I'm saying is that action to suffices to show that the use of a crack points and by some fairly simple what elementary observation singing it helps to notice that these lines actually go through the origin which you'd never guess from the way that the drop but they should go through the origin and similarly the fly the horizontal drilling area that's exactly just an argument that was used by them to adjust when a show that so now we know
that's the twiddle at the end after all see and because of this that you can just lift the McDuff slinking so you should think of what we wanted to do is 1 of fine obstruction to finding a better and better we find structure To that end that From ends well if you're so when you make that point 1 if you're not used to think about this important having problems while I wanna make the following point which is that generally speaking that obstruction should come from some sort of polymorphic I mean of course there are other obstructions is inflected combating so that a little more variation on nature there these Eklund Hovard bastards but it's fair to say that it's reasonable look for a home off a curve that somehow obstructing something better OK so now it is 1 of minor right and this is where we use behind :colon an argument so you should think of the curb it's gonna live in the completion of some could borders on of high-dimensional in this case it can really have any new human dimension His we want to find some homework occurred living in a completed some plucked to an end so the hind current idea is to find a curve in the former national borders on that lift nicely to higher dimensions so that's a little made and I'll fill in some of the details later in the talks but this is where the ch comes so the way you should think about it is go back to McDuff and slight Of course there there this McDuff like statement that these are points on the graph is also an obstructive state the obstruction components that you can't do any better so the idea is flying home off a curb that's giving me stuff and slang four-dimensional obstruction and then well this thing it is like a product so roughly speaking we wanted is you wanna take that home off a curb instead of mapping it into the completion of the ellipsoid into the completion of the ellipsoid cross-party when by say sending it 0 and so meant to finance home off a curb on 1 using so that's the general and yet this this this kind of essentially behind from an argument with the siege and easy to use them any questions about the you news of this is this discussion about how as you can see these lines go through the origin and and so but there there's so and Felix have this nice argument where well these these ellipsoid embeddings space satisfied some basic accident means of for example the monotonic like this function has to be a priori monotonic lean on decreasing and it also has to satisfy a sub scaling property and meaning once they have some point on the function if I draw the line through the origin connecting making point of functionaries has to stay below that line can go above the line so using those 2 our points together with the McDuff enshrined information you can recover the whole and this is kind of a version of an argument that was used by me and like that's it's not there's no sort of interesting homework curved just 1st principles in OK so now may be a little bit more explicit about what I mean then to buy 5 however she so we 1st want to give a brief crash course on embedded contact Cincinnati a very brief crash course hopefully more of the cost and crash standing the motivation is 1 years embedded technology defined now we have are easier to crash please abandon that the kind of the complete opposite of a muni course where everything is done carefully in much detail so for E C H I have a lien manifolds so wiser 3 manifold and land there is not generous Contact defense now there embedded content analogy of this care while there is a variant of Newsday's inflected field In other words well it's the homology the chain ,comma blood and the draw which is generated by certain sets of Ray borders and the differential deed and certain well embedded except there maybe just mostly embedded service in our crosswalk so I women doing
so much with them with contact knowledge is this kind of thing at this conference that I think I just wanna drier pictures of the king she should have in mind is really kind of curves the gate count and well the kind of cars they can count on you should maybe just have a picture like this by which I mean to illustrate the following summer cursed can have genus they can have are trailing many positive punches the number arbitral many negative punches that's why I'm saying that the chain complex should be generated by sets of rate the year cylindrical contact analogy you just have a cylinders 1 better than 1 of them a whole set of Orbitz here but we don't see 1 account of all such curves who wanted single out so embedded curves in some clever way and we and up singling them out by this thing called the EC so we count so specifically we can and BCH 1 on the Net somehow forces the proves to be essentially embedded what I knew anything most that is a component like this actually has to be embedded but if I have a trivial cylinder here this this could be more bleak and so CCH announces it is a great thing this is all due to To Hutchings of course and will actually almost done with the crash course but you might ask why we doing well 1 reason is that embedded curves at a very nice thing and certainly that kind of transfer salary issues for these embedded curves are much more mild because they can't be multiple e-commerce but another reason which is a bit cooler is that this embedded content to Margie well looks very much like Hall Norfolk proof theory but it ends up being isomorphic to engage so talents has shown that is canonically isomorphic 2 of sided with law this start here is just some grading business which we don't really want to talk about so this is the same of admitting clerk or and why I wanna say much about this but this is the gage theory it also only depends on the 3 manifold y and not on the so somehow embedded content analogies allows you to encode topological information about the 3 manifold coming from gage theory into information about the contact and some plotted geometry of 3 manifold associated forming so it's not announces the end of our crash course but it for our purposes actually what we care about is that we can use this isomorphism to define waters we can use this to define the word is announced and by which I mean well if there's 1 thing that should probably clear from this summer school you know if you wanted a find Kabore maps on some version of some pointed field theory normally that gets extremely here I'm however sided with flair ,comma allergy does admit the words maps and this has worked out a nice detail by comparing rocker and so we can use the fact that Whitten theory admits words maps to get with the board's a map on embedded content knowledge now a period that borders and that then would count some kind of cyber when Monfils but hutchingson tabs have shown that these were the map satisfies a nice actually the looking particularly they satisfy a home-office acts in which will be very useful for and in particular the home currency Bloomberg as I said before that we won uses defined some nice four-dimensional curves and that's the idea we're going do circuit board isn't that inhabit the border the map for you apply the Hudgins tells home occurred axiom to get a home off a and then we're going to pay very close attention to it to make sure that we can lift correct the fastest way the know I will not there in usage capacities interestingly yeah you can ask me later I don't refuse ever talked and interestingly we don't actually have to know much about the stage capacities to do any of them and this is maybe Michael recently can you put in writing which is that you probably should really think about the curves themselves and not so much these capacities which user-friendly but for a very delicate applications the lose information were much more interested in the Kurds themselves the nevertheless inhabited by the way this sum collected to Borderland I could write more about what I wanted to be I think this setting actually wanted to be Volume I think I want to be exact selected reporters of actually once the weekly exact but I think I'm gonna skip those details you can ask me later OK that was a very remorseful I haven't told you so they're here this is part of our crash course studies the Agency anytime you have an exact inflicted the border zone between contact manifolds you get induced nap on embedded content not now have to tell you how to induce that borders and the purposes of our problem and saying you would think that you want exact aboard but are wars was actually what's called weekly exact which is fine for what we want to do but anyways the whole and more last year yes at a later moment so I'm now going to say this is not just general facts about ACH I'm now going to induce for you the borders and that will be relevant so I know when I go back to the group OK so someone produce some sort of syntactic borders and and while the general game here which is certainly familiar if you've done this before is that the water them should come from and so far I the McDuff enslaved computations McDuff punchlines but we know that I can in bed well and Calypso because stretched by a factor of the handsome embodies the ends these were ratios of products to minority numbers that are authorized to 2nd that this year and
I actually want add a little Absalon for sort of technical reasons that you can disregard at 1st and then I wanna embedded into the interior of well and is the right this isn't easy the prospects along and where this CD is equal to the BN plus a plunge Whittle so remember that CBS was itself and and man so this there's a lot of fun here and they're put in here because of the convention I learned this phrase for parallel option they're putting here because of the convention that statement should be correct instead of approximately correct but you should maybe ignore them at 1st reading so the reason or putting these apps here's remember we wanted and not degenerate contact form and if I took a ball here you don't have a more spot contracts so you really want to think of this as a kind of ball but I'm just perturbing it a little bit you rational and similarly the and it's a rational number someone returned this year rational and and also there was a point that made and I were talking about that I can't actually and that into the Y 1 embedded in the interior for what's coming so I might have to make see a little bit bigger just and that's if you're wearing that this for the 1st time you truly can just ignore all the other so underwriter picture so this is the kind of picture that you should have in mind is and the see the bus 1 and my ellipsoid is just embedded in here in some strange way there isn't you 1 the end of a long so now I can write down toward is so now they're removed well outside of the interior 1 billion-plus Epsilon OK to get it borders on the backs of the poor 1 yes that's right so this Gujarat weekly exacting practically waters of the city should think of the comport isn't as this thing act the United Daughters of facts it is from the interior of the school of thought that the whole of the in the interior of the ball movement get worse now my homework occurs again set in the completion of acts so want complete this by adding cylindrical yes to it I see the issue that this a simplex demanding that might not be exactly what to do so when a complete acts by adding cylindrical lands OK so again weekly that has been not import I'm just saying we while in inexact selecting embedding you would like your synthetic form I'm here to be D A 1 form that restricts the boundary is the relevant content but we did not give an exact simply to combatants we can't necessarily find however always actually need is we need this inflected form to be exact because we want to rule out some kind of bubbling for example and we want to have we won its restriction to the boundary to be D of the relevant to define borders but I assume most people are more used to being that exact symbolic border so so you don't lose much my guess of his right to both sides so so this isn't 1 of them would too the components and I want add a cylindrical and founded so what acts bar looks like is acts and then I had a cylindrical and to the top so I added a 0 infinity Cross of boundary of the ball I mean it's actually sleep see and I add that on on the boundary of the ball and all and on negative and on the boundary of the ellipsoid mail so take this thing is at an end to the top and an end to the boss that was a pretty standard in the story object to the current lives in 4 OK so they sit still are curve will live in it's hard In a moment of the curve for you but let me answer vexed question and tell you a little bit about what the home off a curb accidents at from it that produces the war an accident attaching it to the it's far OK so now remember ECG the water so because the CH in Minsk enormous amounts I didn't side twiddle his induced by side mapping well the embedded contact analogy Of the boundary but this fall and 2 the embedded contact mileage of the boundary the elusive OK 1 now and it will satisfy the home off agreements but so now 1st you wanted very easy calculation it's an easy calculation while so this easy calculation involves the following I wanted tell you a
few things about the embedded content knowledge Of course they haven't really define embedded Conte acknowledging any rigor but remember there's some Ray boarded ball so I wanna tell you what the the boundary
of this year rational explained while it has to rate were and to indebted reboard lets you know I think of anything channel might Mohammed and take the final the thing that seems to be a popular strategy this innocence but I'll take it well maybe I should stop these I'll take a few more yeah so the states to do that but I just want to wrap up at a reasonable place unarguable up 1 but I I promise I'll be able to wrap so so this has to embedded orbits and carry on which a musical Alpha 1 In offered to while the action could offer 1 agency the the action of to is the plus excellence well but yeah I apologize for you really do I just want to finish general trend and so well the boundary of the other guys similarly has to work In 1 my mind the then debate 1 is short mandated to is the 1 so anyway is the point is you can show and that there exist you can show that might now the trial what actually remember any stage Wojtek word sets so if I take this Corbett Alford too repairing on and take Jeanne plus 1 copies of it I can show that back scrapped to Baidoa 1 OK with GM plus 2 copies and so on now that Homeworth occurred axiom the point of all this is that the whole market axiom now says that there is some broken some possibly broken home where Victorville the 1st Alford too to the G plus 1 2 the 1 to the G and forced to repair I promise I really am almost done and so just to emphasize the next question so what the homomorphic relaxing as saying here is because I know that the war is in that sense this element of the siege to the summit of the siege by know that there's some for the home market correction says that there's some homework occurred but it could have multiple levels here you can show however that in this particular case partially because of the nice properties of the Fibonacci numbers this curve has exactly 1 level can show in this case then there is exactly 1 level actually write down the curve so the Kurds looks like Willenhall driveway so that the current looks like this so there's Jeanne plus 1 Poseidon and there simply covered and 1 negative which is a GM plus fold cover you should think of this as the "quotation mark Fibonacci Kerr OK so now the say the point then we can go eat lunch but I do want you to have this picture in mind this is a very nice another point of these curves if you stare at the index formula so these are index 0 about sort of just a general thing that happened was words about story but the point is if you stare at the these current and apply a Fibonacci identity you'll see that when I lift these curves essentially by just crossing with ardor that you and then they still have index 0 In principle when I put it in a different model is based index could jump that I write down the model this correctly he said index 0 involvement and so we can use them to do the hind common argument what is behind her managment while regret I did and I want to show that I can't I was so that I can do better than the product and so what I do is I take some random betting which might not be a product on and then I can isotopic To the product and if I allow myself to shrink the ellipsoid and so by doing that I can get a 1 family of almost complex structures and try to move this curve in the normal way so I just have to prove the compactness results showing that these curves actually persist and I also have to do some delegates signed counting to make sure that you know there were 2 of them to stop being do all that and what you find is that even if I if I had some non product inventing this curve is still willing to to to obstruct so so anyways so you have to kind of do this compactness calculation and also assigned counting but the picture you should have in mind is really this picture because is a sort of nice the minister so if Nunavut apologized me


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