Holomorphic disks and special Lagrangians

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Holomorphic disks and special Lagrangians
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Special Lagrangians in Calabi-Yau manifolds are expected to be plentiful. However, in practice, it is difficult to find special Lagrangian submanifolds in compact Calabi-Yau manifolds except for two special classes: fixed points of anti-symplectic involutions and holomorphic Lagrangians in hyper-Kahler manifolds. Thus it is natural to look for a modified special Lagrangian condition that reduces to the standard one in those two cases. I will describe how moduli spaces of holomorphic disks give rise to such a modification of the special Lagrangian condition.
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so the a 2 2 thank you for the invitation to speak what so I figured I want to start with some motivation for what to talk about some of the so the setting is generally Columbia manifold because the notion of a miracle IBM would be of little things thanks the manifold dimension and the Commission complex mentioned after you consider that the information supplied by the former trade as a compatible complex structure so say complex from integral the Desmond's that can structure and capital is a J Home Office of well the General Office 0 4 such that popular constant you take the wedges capital again complemented by a little later and and hands the meaning thing this visibility says that I sense that normal capital makers want to genes so it's not magic and that this condition good sense the tension chief of she flatly armature so this study on maybe this constant is chosen to look normal males the 1 the so have to be abundance of natural well In fact the hurricane losses just that so ,comma 107 and then implemented
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part of my statement that the time for about which is the same thing as this is not trivial but this list and approved its for the idea that the government minister the value of some of the those little London's interesting for many different perspectives that wasn't that interesting differential equation or disapprove and flies the minimal my minimizing the loss equation going to so actually yesterday among I think so it's not compact you can still make sense of this by taking Compaq was supported the formation of the like that so I I'm going to want to usually has a mix of compact but None of not always was certainly examples all right so that special margins so why years many reasons but I don't think so originally made me a lot of tensions run special London by the Swazi conjecture other people talk about that I'm and I think maybe special bonds are not really so important to the size funding for maybe the most interesting part of that so far but for me there's ball the object of a kind that acts as a new meanwhile stable what you want but you want to say something like that it's not going to be true but what if its so and I mean what is that I mean what's what the reason why would you want to know what stability is our own in for example you might want to try count special abundance of manifolds or something like that so I something of a special function of the representing some classes then commanded acts should something like like a mirror to but now don't come and on the other variety of individuals stability is something 1 to be interested in and then I mean there's many competing White met but would you might want to have a some some sort of emotional stability conditions like defined as vigilant which reconstructed by displaying an object to be stable surmise there will always be fun and stable if it has a special margin represented but I think you will see on you Well I mean they're inspired by it but just you can't actually constructed to stability it is a volume in the it was 1 baby the net well into intuition for regions abilities come from but actually proved there exists and we value all the examples especially on the way in all of this has had far far fewer than you need to construct the stability conditions and were now was the moment for the country I'm sorry he right I'm saying the intuition for against spiritually it was in physics papers for the name of the but the Suns under performance items under and then visual and build on that side Douglas S. Michael Peca on in the key he thought about social London looking at certain examples we construct many of them and and in non-contact case and then tried Mary that some kind of comical like categorical language and then Bridgeland found he could construct such things on categories of coherent she's but the actual original setting of stable stable it's a special dungeons you can't really actually rigorously constructed any stability conditions this way the 2nd is to move the party has no idea of the saying that this we think the launching of the bank president of 1 of the century OK 10 that's 1 way of thinking about the merits foreigners and 100 decomposition of objects in the back of the of In other words in some sense like stable objects should be you know in some sense generic or something like that it's like you can you compose a matrix and is Jordan normal for him and his call charge functions integration of the volume implied that the agency was something about In a statement from people there's not really a notion of some object in trying categories sold them and that was kind of the beginning of Douglas's I'm working on it possibilities and and he wasn't it was inspired by certain examples was exempted right down the non-combat case of special lounges but in fact as so far as I know you can't really actually defines the conditions and stay on combat flabby manifold using special London so example
yeah and I was kind of trying to get my motivation in terms of things which you know know I can do know that I think so I wouldn't approve Mohammed said as the special dungeons our for somehow plentiful the man who goes by the generic and something like that analyzable matrixes generic among only also said that had to reentry the the anymore the woman with the word now and so on so example 1 is to take some of them and make a day of the bombing of so many men 8 a man so map which is and that's about it city of information In some sort of like a souped-up version of and the Home Office OK so many why this is a useful thing the moment you have this you get well the combination of these 2 riders this implies that this is applicable law the moment that the 56 points are of dimension 10 really mentioned and think about public contrition and then again the occasion of the mansion and this implies that when restrict magnitude of 6 . you get 0 0 which means that the fixed points on the project and so on there again using this to get there when you strip Omega to 6 points you get so has been able to Woods sorry should concentration so that equal to its company conjugate that means those imaginary part 0 the this is it means that fixed points on special around is I don't have this kind of answers about the candle was the conclusion of funny and manifold and then so summary of the fixed line of the are especially London cited examples maybe some examples just for concrete has been constantly and national Euclidean space companies and ensure including space and even the Omega the status of the reforms are to the and this is the standard whole logic and form and then you can take to become the foundation has presented itself positions all 6 points on the far end which especially Rogers OK so that's the example any of them the contact cases so in this way just write down this year I didn't know about the existence of an home of a cumbersome but the conclusion but this 1 is a little bit harder to get on by and say take extra me the that's good .period CP and such that he could see 0 sorry and plus 1 and P here is a homogeneous ,comma nominal degree and to that so basically the Columbia out feral says there exists mail capital make such that this act with induced J From crumpled complex that the space and that was Columbia and in fact moreover this study out there in the unit as part of it implies that I'm free but can be taken to be discovered conjugation president will those metrics that I mentioned to you the other side images polynomial With real cooperation thank you all right so there then you get but the real part essentially that of some of them had
reserves of Americans visited India special wanted manifold OK and that examples that example 1 example those assigned to special parts of the basically a docking about glad we are mindful of some kind of company conjugation generalization there In physics is just real hot the real part is special wanted so the 2nd examples when acts the pitch where the paper tariff structure Somerset such a quintuple it's high it 2 I play the Bulls JI this is the 1st idea became part of the structures and I just put my ministry calls came so this is essentially a Pretoria relations and treatment soft and and also we want say that on I it is going to be just GE suspected being a 2 4 defined by G. I and performance Jenny he Jr on Medicaid care so all 3 of these forms have to be synthetic form so that means it's closed no agenda both others not generate but that's "quotation mark to use telematics for each 1 of these 3 different public structures To begin with the same Gazette that compete with 1 another the 1st thing you thing you things don't exist the so there is an example of course just like alternative in space and when will covers subject some of which onions with themselves they can also do so so so members and news was kind of a bad example shows how you can take but acts areas example 1 meeting with people to his feet so this and then you can construct the height Keller a structure in the lobby of all right so the money will go mission we want out of him so you have then you can decide 5 Apple flies going to be the sum of on May I needed "quotation mark woman it's going to be it's I Moffat to 0 4 the type that I mammal acts home of the branch in the city have like so a complex of manifold with respect to this is called a halt to its I expect it to them pensions so then some pretty simple thing to check is that if you have a whole lot of 1 the so for that say again Newman inside you can have the capital murder which the following I in the eyes -minus scrutinize one-time semester came and went to the power and to officials said here the dimension of acts state but have been reported to them otherwise you can't possibly have the thing so it makes sense to divide by 2 assaults holes and actually there sir if we take such a thing that a major Jr J and capital made is cloudier so using hydrogen structure reconstruct Ecolab Bealle structure something is less than it was at a furious less so given hydrogen implies "quotation mark and insulin the fact that the sales that if Obama's wholeheartedly and then it is special and general spends too but with for this Calabi-Yau structure the most like a lot of extra publication actually it makes life a lot easier to find special vengeance because but this 1 I'm going to get started you know I hear of a thank you gonna die I began of to repeated that know of reach not to reach its Obama's I commentary that our the help so mean looking at the the
man for all the way In her so so example you can take the elliptical I told Federation the dungeon on the TV service from mentioned above on and me is that this is the year by the algebraic geometry the high-dimensional down too so once you have regattas setting you struck a lot of special badges despite the average allergy constructing say but some homomorphic submanifolds and then looking at them with a different with respect to a different structure with respect to James I at you doing when you look at the are structure that's what I propose rotation and gives you many many special advantages sold History so surprising thing is that it has come back the all example the journal and you can't do this this right maybe I should be in the history of this kind on his side 1st of all you can say 1 thing and 1 the saying that may be we let him know that it is now OK OK there is the opportunity for nation 1 this anyway because of the false in citizens explained McLeod proves that that you have 1 special plunging you can find a lot of nearby special dungeons solos not only model on the 2 vector space H 1 of so the question is I'm in another and that's why the conjecture I would say should have a special about and vibrational lowered Brennan many social margins even once a mention odd attempted workout it and if renewal Mohammad's example then we should really be able to compose any object guy category in terms of special Branch some so-so special underdeveloped generic at me this is definitely not generic conditions and so on have yeah you this In the future it well I think about warrants it immediately all right I mean there's something kind of different between them I would guess I would just not at all it was 4th so where what happened all the special branch so many specialized condition is only an approximation to the real thing and as usual but the Daventry if something is only approximation of the corrected by or modified something to do with it hold the whole of last year was grateful for all right so why does the scene a reasonable thing to to expect the 1st of all it's not hard to prove forgivable bit evidence for this hypothesis to the future well basically some of the patients so that they act Cheney had propelled by the right arm president instead of in a closed Major the came as a great place to work with 1 and so this is a complex structure of each and we just use algebra the circle of public structures so for almost all for all but time a number of public structures and IRA there are the whole mostly stable maps engineers the it's so that no 1 was home office since the project there are no such guidance isn't so that means that has sense the major reason that special the whole article Longines special and you had many of them is because whatever it is that's modifying the special bond condition isn't there In the whole of the Rongen case because there are no homomorphic maps that are no all of maps for know almost any news complex structures this year my home office essentially so we don't know that for mean there's no Baltic Marcelino ashore you know a sequence of public gestures are virtually close to there and all
of maps which lies some of which which kind of makes you think that whatever contributions and Jr they should be in some kind of obstruction you know when we get rid of the obstruction when you do obstructing their properly they shouldn't contribute president what is the use of destinations worldwide I was in the otherwise true I mean it's something with calibration it's not a hard argument that it did not the idea and explaining why this tells you something about this idea he because was out of the community and the United legal online proved so but the of his idol In his talk on that all right on the other hand it's not so sold it's not always so straightforward Soviet look at it the the example 1 be again so survivors of Inc the invective space with them and walked 3 now then and it's and the that is the number of the problem to build a properly countersigned zones of the number of now from the experiments the is especially 1 then and the so it's simplistic area the so there and the firm's exponentially the it's a lot of all of the best ending on these 6 points and solutions so just seeing when there are where the going the whole of the disks won't answer questions but so this for the the idea this is against the idea but this is the government they think it will produce which has property which is expressed this year the somehow in the present moment does he just take over the world so hold forwards "quotation mark yes yeah so Sony the special 1 in addition to be modified has something to do with homework forgets listening as well on all of it just be a lot smarter version of dungeons now sometimes there's little under the weather when there are a whole lot of this is suspended against any of looks bad for idea but actually possible that further than just saying and all of a guest in the mess they want right down explicit equation which has to be satisfied but of but so there are 2 of the owner it says that on so they said example 1 being and there's going to be unobstructed With trivial now the all right so in the Navy whatever our modification of a special and unique vision is they should have to do with the bound chains of on all of it is that everything that obstructed a minimum to do if the armed and then we can ask is London unobstructed the understanding of what that means is intuitively trying to scenes of and and then is if it is unobstructed while there's this the truth is that say Howard's unobstructed which was the bounding change and if this damaging is trivial them again you should have no modification to so this sort of this means that more specific at the moment especially mentioned conditions using mountain chain and we think of the company where 0 without any direction exactly they are that's not say said right so now well I would like the defines as they have done this year and 1 some the honor saying I want to find flaws knowledge show me the commander in summer and and gunplay and with respect to certain differential the usual president basically free I think London letters state clearly on 1 so London there's no rain this is the field won as usual
thing the they got there own 1 there this is the missus clear the new on the job equal to the some and of the number of said the strips appropriate and say what you say only integral supporting the and that's the thing that makes some converge and if the sum over you here and with you're cute all right the trouble is that in this generality this doesn't square To do you kind of think the National Fire differential the no 1 wanted to of so in general I would like to say that a the unit would look
at see so are advancing on a long look at the system we like a picture of new squared off the use of some or all arms like this was in this thing is basically a sum over it's boundary some of smugglers face subjects from a the but so it is the only thing that European you really good shape because of the new new Verbier boundary of something would be 0 the dollar thing is you can get pictures like this as well and sitting you can consider splitting into 2 Florida like died and spread into 1 floor trajectory and home-office desk it is the home of repairs insiders the home with them all 0 and so on at 1st order you want to choose some chain number 0 0 look like this will you have a bunch of of disk then 0 so essentially you want to pair them all the combined in some way In the Swanson chain inside of them 0 another a with respect to which chain level fearing that Andrews made different ones once you have this you can cancel out the boundary over there that is counting accounting things which intersect the 0 well I'm not convinced that the drawer but all right so it is in the end what you get is that you can't define floor-mounted general for followed London summer but if you have a bounding you can in fact that definition of floor-mounted hands-on on abounding chain the upshot the information of features that the number 1 depending on founding injured on change is also so so what heard about this so you have an incentive more than that you have then man only running the 1st or have to write something in terms of the ancient structure associated to Obama's 0 in Gunawan and you give us more of a tiny equation and so this is only the 1st quarter thing you want to happen and the so founding change this is the 1 .period and we want to do well as a special advantage the condition on both the Lagrangian and abounding Jr my the condition for the pair the and so now on the side of too much motivation and too little of the Pacific but roughly speaking the way to write some condition or beginning of alliteration condition interdependent abounding founding that the reflects base .period and some of the fire and then sort of look have the Rongen starting at the start the stuff as a be clean some of these people you that you are looked after home would not be present covering space and while giving taken defined my number which doesn't depend on me it all depends on hold on because of the past well I essentially want to look at the imaginary part of America to assume that this is an integral part of America remember starts at 1 0 concerning basic module obstruction the have satisfied then you want to
serve In a great amount matter over the whole of the Union of all bound in chains In this
all of the gamut he's so you have and is government come along and as you Will between numbers of underlined you're bound in chains move along with you because of way construct such things as much as I do so I was kind of surprised the differential form the federal form with what it always lived some degree 1 so and then draw nearer the three-dimensional case something he was being a formed gradual surface inside of my the Dodgers and the Soviets a visa the Sony commanding about things differential forms than I would force after we were virtual dimension of such a thing it actually turns out there I mean so for example is an trying to draw them the dimension and 3 so this is gonna 0 3 Rafael cited his visa which is a two-dimensional surface that I can I consider my family of beauty and as I move it along I have a family of 2 national services out a three-dimensional as a manifold my it's always inside and I decided integrate this imaginary money on and over all those things and so their fear the United States this decision decision to make it theorem is there this this enrolled depends on the end points but the holed up in variant of what the that means that you can for example move 1 points and it will get done "quotation mark undergraduate you get and that gives you an explicit home of against dependent equation involving these capital may go and and you have to add some other terms introduction on other occasions but when you get a sensible were so so called it has been sensitive and some functional depending on certain that hold that the class of government the and agenda look at whether the tragic region contained by varying plus some way possible gentlemen thunderous and this gives you an explicit modification specialization equation which with which recovers the subtle and innovation in both examples journal start thanks the court you want for this classical tunes for the 1st found and in how much easily won her head of the university's was placed on the use of OK a it it's so anyone think about them think the on the 1 for certain that we get is essentially the the differential of is equal to yet the states the bound chains essential to me some kind of song both sorts of UK operations where John the classical part a surplus man so I just look at the Potsdam from Autodesk's loss this part of the UK strictly speaking also get something like the code differential of the it is equal to something like I forget the exact form that something like that sign of the Lagrangian angle it it was all sorts of stuff OK it's a complicated formula to remember offhand but perhaps I should prepared but I can show it to you most of what you really the history of the city I think you 1 thing that I mean physicists told me that I am quantum correcting the entropy I think if the new rules that work the usual special London condition was likely to you on the provisions of the agreement you the real begin addition to handling especially Londoner presented for Hamilton heights the class is all that is because he was not used in this cancer I'm not sure that I would be so I mean I think you have to throw in some singular rise as well but I mean this is sort of in 4 years can you go through surgery with I mean I would expect that you could that I mean that 1st of all we have to find examples of move 1 of these guys which is not trivial and smooth solution to this a non-trivial effect in this 1 what so what look so well what want do you think that you know anything and you don't ask for them it's not a question of whether the lawyers republicans I wanted to do these responsibilities that here on the the future of the president questioning this year and wondered why you picture I just wanted to ask the question but I would like to see in the main body of the talk essentially what happens in that in the case of the Andalucian is also the answers why dilution case is the same as the home of but and you have you have lots of just 1 year on year on your London's are manifold but that they ,comma instead of In pairs like so think of think of the complex combinations of switching the red for the purple dye and now you want to do abounding chains so you have to take this through boundaries of disk and connect them by some kind of surface but they coincide Sears service can be the empty soft so just a minute just the surface which falls between the boundary of the purple on the boundary of the red so it would mean the is the way cotton was saying is that I mean basically you have all these extra operate boundary terms when you look at those who one-dimensional model is bases but they come with opposite sides of the country the wall it's a different side can I had an interest in major ways of looking at these disks and 1 of them gives them 1 side so signed other and those of size of but 1 mark .period on here which is what you name it to them obstruction there it comes up society have no marked points on it which is what we counted then lake on the same side the ministers of alcohol request what would go with


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