Some topics in Lagrangian Floer theory　Kaoru Ono

Added: 2026-05-03

[00:00:13]okay thank you very much for introduction and today maybe I just want to so my what I'm talking about will be quite remote from the time of this conference so therefore I want to make my talk not so boring as much as I can so today I want to talk about this grocery maybe with a focus on three things so after this is actually based on on joint works where is with candy and Young and if I if time permits I may also mention our joint work with Muhammad of Zaid so okay so the story is the following so for instance if I consider this S2 S2 is just a 2 sphere like this and then I consider the height function maybe height maybe here 0 and height one certainly it is a momentum up for the rotation action and then for this we consider some function like p o S2 Y is equal to Y Plus three over Y and this t is actually that t in this Lambda so it's actually indeterminate and for CPN the function p o CPA y1 y n is equal to y1 plus the y n T over y1 here so so this is a function and so I may I will explain a little bit about this library on the floor Theory but somehow in this story case this function somehow knows the plural homology in other words after we consider the critical point of this one and then we can see the where the uh floor Theory so maybe I just start thanks so X Omega via closed that is compact with that boundary simply thick manifold so namely Omega is a closed 192 form so locally as everybody knows locally this Omega is written as a DX I divide I and I run to 1 to n this is a double zero and then I take J it is almost complete structure such that if I Define the bilinear form using this simplexiform and this almost complex structure and then I have a bilinear form and if and then I assume that this is a limit metric namely symmetric and also positive definite so such a thing always exists because the group of simple terms linear simple contains unitary group as a maximum complex subgroup as as you know and so this is a first thing and then secondary I consider maybe two sub manifold so-called lagrangian sub manifold so namely they are just half dimensional sub manifold so that this Omega received to this sub manifold complete advantages so this is a manifold maybe I consider only the closed one so compact without boundary and then the somehow the floor complex so it may not be a complex in general but I just want to tell what it is so maybe I denote by CF and maybe L1 is zero and there we go so this one is a free module generated by intersection point and so the original Locker floor after he considers the free mode over Z over to Z but today I consider everything over this Lambda or maybe Lambda 0.

[00:06:28]so so what they did so this is something like a uh Lauren series but exponent just real numbers so if we if it is the infinite sum then the the exponent diverse to Infinity and then if you consider the case that all the exponents are known negative it gives you something and its unique maximum ideal is this Lambda Plus namely the all the exponents are positive and then it has so-called variation so it's a no archimedium setup so you take this element which is non-zero then some coefficient is non-zero so therefore I take a minimal exponent of it so that gives us some additive for yes okay and so so free module maybe today I only cons sing over maybe today Lambda 0 or Lambda I consider both cases and this Delta somehow count uh Circle movie strips joining two intersection point what does it mean yeah I also have to I plan to use only one Blackboard but somehow it's it's not good so maybe I just use this so namely I consider just a student like R times 0 1 it is actually sitting in the complex plane and this is a real axis and this is a part of the imaginary axis and then I denote this Tau empty and then going to X so I consider nothing like this with several conditions one first one is a cosine in my equation so like I have almost complete structure so therefore we can consider this equation so this is the something like a quotient migration and then the second one is a boundary condition so how I belongs to l i maybe zero one and the third one is this U Tau D and power tends to plus minus infinity then this one is equal to p across points so namely I have some money for decks here and then I have two reference and manifold like this every under one and then I have a cylinder like this and then I consider mapping from here to here like this and then so this part maybe map to this so lower down there is enough to l0 and upper boundary is map to L1 and then if Tau tends to negative infinity and then it converts to the intersection point and then if Tau tends to plus infinity it may convert to another since and so maybe I consider this as a p plus P minus a solution of all such things but then it has obvious symmetry because we have a translation symmetry because this equation was symmetric in Tau variable and so therefore we can shift the entire Direction and then I if I assume that maybe I didn't really say this if I assume that we are they have a skin structure and then I can also give orientation on it and also we have a dimensional formula for this guy so Dimension is actually given by the difference of so-called Mastery as masterovital indices at this point minus one minus one coming from here but anyway so I only consider case that at this moment I I only consider case that this MP Plus T minus is equal to 1 and then the boundary operator is the following so derivative of Q is equal to the sum of mpq and P and but more precisely I may write it in the following way so maybe I take a equivalent because of U from this m p q and then here I take key and then takes T maybe I need some sun here and then I I also have a used area of this polymorphic disk and so this is Delta and so through after he he considered the case of this V2 coefficient case so not this larger case but he shows that this Delta square is equal to zero if I turn off X Y is equal to zero but in general this Delta compose itself is non-zero so I can give you a very easy example that is you have just a line here and then I have a small circle here like this and then I have to click a point a intersection point and P and Q and then I have one orbit like this one slip like this and also I have another axis and so starting from P going to Q with some with some decoration and then Q back to p with some decoration so therefore in fact in this case Delta Delta p is equal to the P but P of the area of this the area of of this guy this thing so so it means that some sort of disk is obstructing of this guy so what we considered is the following things so if you like maybe you can also put it here I think well maybe for us anywhere thank you and so what we did in the following so formulate obstruction to this thing in terms of this 5050 algebra of this fi and I I I I will explain it now and then the second thing is that when and how can we livedify Delta so we we add some friction terms to this operator so that the Delta Square equal to Z so here the keyword is bounding good thing or maybe even weak bonding cool thing and so in other words if we model accountant element and also we have for the construction Supermarket information maybe I may speak about it so three foreign things so in our case after we consider some Vector space maybe in this case maybe you forget upgrading but lift a space over the ground field and then I consider some module which is different in this way and then the filter if algebra starts from B is the correction of multilinear operators foreign [Music] in fact this m0 is somehow responsible for this kind of homomorphic disks and then so I I want you to light this but uh very hard so like M1 maybe I I don't really like everything but maybe I write something and uh oh oh sorry ah this one is equal to zero and if I consider this thing so it's a binary operator and then apply this in one and then suddenly we have time like M2 M1 C1 C2 plus minus M2 C1 and M1 C2 if it is equal to zero it looks like likely true but in fact we have more terms like M3 m0 and C1 C2 plus minus M3 C1 m0 C2 and one more time namely m0 is the last one so this one is equal to zero yes so it is a kind of rightness rule but with these terms and then we have M1 M3 plus so this last one is somehow if m0 is not present then it is a sensitivity for a multi operator up to M1 boundary so so last one is somehow I don't dare you [Music] so if if everyone is good sorry M 0 equal to zero then it's accessibility and then I have an infinite many such relations so this is the filter without a loop and how can we Define such a thing before slots I think so so take any relative homozy class so like Aries lagrandan exit ambient space and then consider polymorphic disks in this class so that we denote by M K plus one beta so in fact here I have to consider so-called stable compatification stable enough we have to consider modular space of stable Maps both as well Maps but somehow it's it's just abusive notation so no abusive motion so stable enough compensation of moduli of holomorphic disks with K plus 1 boundary Mark point what does it mean so I consider some disk say and then I have a boundary my points like this and then map to here I have L and then I have must be like this so this is a this is a uh not in the view here and then over here I have a bunch of vibration marks namely if I help you and then I can see that U of CI and this gives us Elementary l so this is this evolution so I have a K plus 1 abrasion Maps so namely for each at each these things and so for instance if I take this V for the drop complex of this F well more precisely I have to shift degrees but maybe I will omit this but anyway so I take here from C1 CK and then I consider MK beta of PSI one double x i k is equal to uh evaluation zero click kind of integration along fiber and then everyone's got C1 if it is the CK so namely I pull back differential forms from here and then somehow pushed out the differential form to this and here this space may be very ugly space so therefore to make sense of this we need so-called virtual technique which came the sky and I developed maybe mid 90s and then further developed by uh yeah our other locks and so this one is also integration along fiber certainly if it is not submerged on usually you cannot Define it but somehow in this technique we thicken the space and then make it submersive and if you take thickening then you have a kind of normal bundle and then you take a tomb Cross of that and then somehow you read with the tone cross and then take integration of five or something and then we can actually make sense of this so this is a MK beta operation and then our MK is equal to the sum of MK beta and then t to the part X power the area of the how much disks and this summation takes over beta and if we don't have this it may be even some so therefore the sum may not make sense I cannot make sense but we take this one so-called group of compactness somehow uh guarantees it converges inside the tea addict topology so so this is our MK and then this one satisfies this sort of uh equations so simply because if you consider disk something I have to also consider mapping like this but then if you look at the limit so limited somehow looks like this right maybe this is zero and this so so the limit maybe maybe like this right so the boundary of this guy is kind of this and then this part is already some air operation so now Bob's single k0 and on K1 then MK1 operation and then now it's the input and then I have a bunch of another K2 Main boundary Mark point and then this is actually MK2 operation so therefore we have such a quadratic variations so so this this is a reason why we have such a suggesting okay so now uh okay so now I tell you a little bit more so what is the bounding change or bounding two things so B maybe I take it from h I don't know sorry sorry maybe I have to say this so originally our MK maybe in that case after it is uh defined On The Run complex of air okay so it's lambda zero and that's okay to to this same thing but so you you have such kind of things and then actually we can also using some algebra kind of homological algebra actually we can also cook up some operation on the homology of cohomology of this guy so this construction we call the canonical model of this guy okay so this one is also filtered Infinity so if you have suggestion in chain level then we can also introduce squeeze the structure to there and then I take B from maybe in this case it's odd of L answer lambda zero and then consider MK B of PSI 1 that is like PSI k so I think this this this this this this formula should be kept so so this one is obtained the following so so M K plus l0 l k and I insert or bitterly number of bees like this and so it is also even sum so this l081 educate maybe anything so this infinite sound so it is easy to consider the case that this has a positive and that maximum idea and then if you have so many a B's then the exponent T getting bigger and bigger so therefore it's very easy to to see that it converges into your topology and but we can also do interesting so namely this one is equal to c times Lambda Plus and for this part if I only consider H1 so not the whole odd part but each one and then we can twist it by local systems maybe I I don't have time to do this but we can put some local system over lover and some manifold and then we can reach the whole constructions using hormonal of that and so therefore we can handle this and if you consider higher degreasings then some sort of Dimension counting argument also help us to Define this but anyways this this is the one Unbound including element is the following [Applause] so this one is m0 of B is equal to Z so because because if you consider the DTA version of this then that may be DX I plus x i x i equal to 0 something like this so look like the equation so therefore yeah it is that one and then we call it is weak if it is not zero but it belongs to the Lambda class times the unit here so this is unit okay so so this is this is the one what is good is that in these cases so like if I I I will introduce this since then I can put B everywhere so everywhere I can put B here and so if m0b is equal to zero then we don't have this term so therefore it has satisfy some lightness group and if m0b is equal to zero then we have a sensitivity after homotopia so so so so in this case it's very easy and also if this proposal to the unit so I don't really explain the notes on the unit in this infinity algebra but the same so if you have a unit somehow some translation happens so therefore we have the same same thing so so in this case actually we have m1b m1b is equal to zero foreign is associative product up to m100p maybe maybe okay so this is the case and so here I say that this is a foreign right so but I consider only the case that CF and add error so the same thing it's kind of self-interception philosophy or maybe but generally we can consider both the most type plural Theory and then here if I take this m1b then from here after we can Define self-intersection through a series it's a self-industration because M1 B squared is equal to zero so this one so if you move one of them so it's a set of intersection so therefore it will be led and if you move one of them by so-called Hometown isotopy and then we have a notion of not only future Infinity by modules and using that language somehow we can also show that this uh so if I if I deform it by hamiltonian perturbation then the floor Theory uh become isomorphic if we take this coefficient Lambda so lambda zero uh it is not really isomorphic but going to Lambda then there isomorphic so that that seems okay okay so now welcome to the toilet case until the case I can I want to compute something so I have still uh 25 minutes or something so maybe before this I I want to introduce a notion potential function Al function maybe this very strange notation but maybe we use this peel and this is a function defined on the sort of Greek bonding with equality element and then here I don't really want to explain anything some sort of gauge equivalence and this our function po is defined over here or maybe even plots so how so this is the following so I said that m0b is proportional to unit that the definition for the weak molecular element that coefficient is this function so so this is a definition so weak motor accountant element is the one so that m0b is proportional to the unit that proportionally constant is the value of the potential function at that point and so if you have to log down some money food and then if we have two that weak multiple elements and then we can somehow therefore the original floors boundary operator using two weak model total elements and then this Delta Square is not necessarily zero but that is equal to the difference of potential functions difference but difference of potential functions so somehow if we fix one of them we get so-called Matrix factorizations by our Construction but anyway so so this is so now I consider Tory case and then I want to come back to this so in this case it's a good thing it's not it has symmetry and so so far I don't say anything about symmetry but here in this case we have a maybe option of the compact Taurus only Simplex manifold maybe conversation and then we have so-called momentum up to the duo of this real algebra and sometimes I I identify this with the sometimes I identify it with the cosmology of the Taurus with real coefficient and certainly it contains H1 of T of Z it's a lattice there and so now so here after I use we use Pi for the momentum up it's very student notation people often use mu for this momentum not but mu is kept for so-called muscle free index in this kind of stuff so therefore we couldn't really use mu here so therefore USP are pi and then I Define the image of the momentum up so this is a put it out associated with this converter okay so now I take you from the interior of the polytop like like this one and then the level said either and some night food so that is our review so this is LaGrange and toras and because because it's kind of five of this so maybe we often say LaGrange and Taurus fiber but also this is LaGrange them Taurus of it [Applause] and so Taurus asked their uh simply transitively so we can identify with these things and now we have a eight star of this l so this is a Taurus involiant differential forms space of porous in body and differential forms and certain it also are algebra and this is in this direction certain it is identified with the ordinary cohomology this one and so in general we consider maybe some drawn complex but because in this situation if I consider this m of K plus 1.

[00:40:03]of this beta right and beta is H2 of X review G so so this is actually Taurus orbit and therefore the Taurus apps here and therefore every construction like product by equation map will push of what by Evolution map of every single fat equivalent so therefore we can work on this level so in this case actually we don't even need to do these Hardware process from this code chain level to cool homology we can directly Define uh I'll open it up on this space okay and also we can we can say that for any B inside of its 81 of Lu so again using this kind of symmetry so we can show that this one is always weak bounding code lengths with moral cotton element so I need a little time to explain this so therefore today I just omit but it's not so so hard to see why it gives us such with amount of cotton element so in general we don't know how to find weak multicult element but in this situation actually we can get lots of lots of with bonding coatings this one is no equivalent thank you this is no equivalent so you just use the Symmetry but yeah thank you okay so we have this operator and so so in this situation uh we have this m1b which is a differential and M1 B and M to B certified license through and m2b associative up to MB model okay so now I consider the following things so B maybe I just write it as a certainly maybe I take a basis sorry ah sorry maybe I should write here so net e i be a basis of maybe eight one over every U of Z and then I consider B in that H1 so therefore it can be written as x i and e i line from 1 to n and the X I just run the zero and then I consider the function like Po or B maybe at U and then I compute it derivative so B is this guy so I plug in B here and then difference it in this but what is this so this is equal to ddxi and this one so sorry sorry maybe better to write one more thing so so one more thing is just this one value okay and then this is equal to ddxi and then I say that this is actually m zero B right so I want to Define I want to compute this but m0b R is that one so in the top of the Blackboard I wrote this mkb if k equal to 0 then I have no size and so therefore I only have B's so that means this is after d d x i and then sum of M K and B B like this and then I differentiate in the direction of EI right certainly this one is multilinear so therefore this one is equal to the sum over K and sum over J equal round from 1 to K of m k and B somewhere and then e i and b b maybe this is this place and then if you look at this compare with this formula this m casings now K is equal to 1. you have k equal to 1 and then you have only one PSI let's start all B's so exactly this one so therefore this one is equal to m1b of this ER okay so I said that maybe we just look at some critical point of this guy so if it is a critical point in a certain sense and then it won't be bonuses so B is a critical point [Music] here then so after this if and only if m1b is equal to zero on this it was of every U h but now remember that our m1b satisfies lightning through and so we have also likely to do Maybe associativity after homotopia and now I use ramula data that is if I consider who homology of ordinary homology of this area and this is after the Taurus so this is actually generated by 81 by top product as another error it is generated by 81 and our m2b is somehow equal to the up to sign it is after Capital locked coming from the question mark somehow module maybe Lambda plus times so so using this somehow if m1b done in season 81 we can automatically say that the m1b bonuses on the whole stream so using this thing actually we can actually show that this m1b is equal to zero but m1b is the operator acting on this uh homology of Lu so therefore this one is same as this HF of this earlier me is isomorphic to the porous maybe here I take Lambda 0 coefficient then I need to take a tensile product with another zero and if you take Lambda coefficient you can see so so so so therefore finding a critical point of this function we can see which you I mean so in this situation so in this situation U is chosen so fast you've chosen and then ask whether you have a critical point or not and in this case this is in this form right maybe I have to do a little bit more [Music] so just next thing is that this yes in fact maybe to write it precisely maybe it may be a bit difficult but not to be difficult but I started from this x Omega J this is a contact story kind of money food and I have five star and then p is inside here and then U is foreign ok so so now I have this potential function of Lu but actually we can have a potential function of x and which is defined on this let me explain this so this V this V is this one [Music] but now I consider H1 of a review sorry sorry uh it's one of the Taurus okay it's one of the tourists and here I have a Lambda minus zero okay and then I have a just a linear extension of this guy so then I get it to H1 of the chorus of r and then this is certain things this part and here I have p here I have the Intel p so on this one is in fact Lambda minus zero to our end okay so this one is contained here so okay so so I have this function maybe I I can I can explain it with that one example and uh okay so let you equal to sum of UI over here right so this is the element in this P Cycle and then therefore it is inside the P star and so I have such a description and so what is the relation between these two functions p o l u of y1 y n and this one is equal to I thought maybe I forgot to say one thing but maybe let me let me do it first okay y1 okay you want while n t Union so so the statement is that the statement for green is a full contactory manifold there is a function like this so that for each U for e is U we have this potential function of U and that is actually beaten in this form so this is just one function here and then instead of y1 to y n we plug in them here and then so in this case if this level is used and then potential function of a view is equal to Y times t u plus T 1 minus U over y because this this area is U and this area is 1 minus U so this is so therefore we have this so now let me put it like a new why and then the sum is equal to Y Nu Plus T over y so so this this new variable this is our potential function on S2 so this one okay okay so so somehow all the all the potential function for larger than plus fiber can be described by using just one function pox but using this and then I also just make one so maybe five minutes okay so okay so now I constant critical point of this PO X let's consider y here and the Y may be y1 or something and then UI is equal to the V of Y and let's also write it as U1 dot u n and then b y is after a logarithm of a y one T minus U1 logarithm over y n times T minus u n or something so then of L of u y is isomorphic to the whole movie of The Solar so this is another statement so it's just restating the defeating the the statement there but here actually I have only yeah one function so just a function associated with the atoric manifold and important thing is that the Y below should belong to this B Vector inverse Key Circle so that is important and then you take variation of it then you can see which lagrangian is actually of non-banishing fluid Theory okay so this is a critical point Theory and I have no time to say about bug deformations but if I say a little bit the product deformation is actually so using ambient Cycles with Lambda will coefficient we can deform these MK things so maybe side group maybe didn't have both D and then this thing is mkb or something like this and again this H1 of you over lambda zero this contains this weak model cotton space and we have potential function so hence hence we have a potential function so so so now I consider the following things so I have a homology of the ambient space zero coefficient and then map it to focused kind of kind of Jacobian ring of this function maybe if I need this B here and they might be to be here maybe maybe I shouldn't really put it no no okay so this is diff but definition is the Forex here you have this a and then I just consider potential function of B and then deform it in the direction of a and then differentiate in this parameter and then take its equivalent clause in this so because Dragon balling somehow you have a lowland series and modeled by the the jacobing idea and here it's more complicated stuff more we need to use some no algae medium stuff but somehow we can consider this one and this is what somehow we call the quota space or not and if I have B and I have B here and in fact important thing is that if we consider the product of this one but maybe we take so-called Quantum hormones with this guy and then this is after isomorphism as an algebra so this is a theorem so fundamental homology can be described also this one and from here uh yeah maybe I have to stop now but from here after we can some uh argument so like a for instance if in this CPN so here we have this function and then the critical point of this guy has a variation which is in this case it's kind of Body Center of the politude so now you have uh log down some money food with numbers in Florida we don't know what it is we consider longer than some manifold which number is in philosophy then it must intercept with the small tank referral tools that result follows from this one and also the result obtained with Mohammad website about split generation Criterion for the sky category but maybe I have to stop now thank you very much [Applause]