A symplectic Khovanov Puzzlebook

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A symplectic Khovanov Puzzlebook
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Smith, Ivan
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Institut des Hautes Études Scientifiques (IHÉS)
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I will discuss aspects of my joint work with Mohammed Abouzaid on symplectic Khovanov cohomology, focussing on open questions.
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Hey if and and to but I think that the opportunity to speak for like the talk may not have any similarity to the title discernible to anybody except I hope that will be true of everything in the OK so let me start with some context not about kimono from all of just about flow theory which we've seen somehow maybe Wessel Lied anticipated that but I have invited my old but you take it
be closed in which case perhaps more monotone is safe you take it to be exact but decided questions on that all are already interesting in situations where there are no differences on the issues of that sort all equally interesting and other issues but it's not the focus of the talks so let's fix of finite collection or closed the grungy and submanifolds at least typically closed down the ways people inside eggs and was so flawed theory somehow starts by saying after picking the compatible almost conflicts from today and perhaps a large amount of all the the that I'm going to suppress we operations Musella From appropriate degree shift acting on the floor complexes which if we assume were allowed a little preservation generated by preview transverse intersection points at these Lagrangian and which counts some in the usual schematic hold polygon With the intersection .period this follows the hands of UK applied to a Tupolev intersections the output intersection is countered with a coefficient is given by counting the reduced hold think polygons whose Lagrangian boundaries mapped to the Lagrangian submanifolds final alive 1 prompted by car and satisfied that all drastic equation which schematically 1 run in this form bands for any given in the form of assigned farm of the different ways that you can use 2 of them EU-wide operations but the top U.S. inputs vanishes and we have these operations save came in the East 1 I'm not interested in curvature particularly to and the simplest of these relations so bad you want composed new 1 managers so we can complexes and we get so far from the differential the next simplistic way she tells fans new to this change so the same as the cohomology and in particular I get in there for now my final collection of grungy and where he is somehow it if the indices is such that there isn't an obvious triangle prompted the matchup correctly that the notation just as you feel said that to be 0 and if there is a well-defined Mutu so here we go are can support of HF L 1 0 consecutive held to L-3 you apply the triangle from but this ring carriers so many of these higher chain operations exactly the the whole thing such a is really sort of the cohomology of Walpole script which is the name the think so if you prefer I would probably not be consistent about these 2 From this notation difference it is quite helpful but the important point is that the higher higher-level operations on so defined by Chang wants him once this X is bigger than turn these operations but just involve new ones and U.S. minus 1 so these further operations that the center cohomology they depend very much on the
particular choice of Jay and all the perturbation data so we don't understand what they are and perhaps the simplest question you could also use when all day in a suitable sense trivial the the equivalence relation for instance ,comma promotion equivalent coming from varying your reservation data moral choice of Jack what was going on in the region all right so the point is that the operations on launching maps so they don't pull logically descend to cohomology but there's not some of the system a purely algebraic mechanism coming from phonological perturbation theory that will let you push the operations of mandatory geometrically defined chain level down to cohomology but once you've done that then no longer geometric the things that the cohomology algebra inherits all actually counting of which makes altogether I know that my game and it I'm not sure why we would get well-defined maps of from these counts of polygons with many sites on the natural boundary structure to The Associated Press In bowl you mean the pair of you the triangle product absolutely so that's here there's a completely well-defined got off to a good case according to local you could try to to count sorry about perhaps for your saying is if I prescribe the conformal structural with many boundary points then I will get a well-defined operations on 1 of the unabated but that's not so the trouble is that it is making that essentially all arbitrary choices you leave the world of the algebraic structures that have been useful in comparisons from all the parts of mathematics to other structures coming up for for instance Anderson tree so both essentially different things because you're counting the region .period immortalized space and you don't have a very formal parameter instead of morally the rigid points in a more July space over a family of complex structures so I don't think that there's some passage between those 2 systems the only reason I think it was what actually hear the composition of the new government of there are a hell so the demand for equivalence relation is to consider suppose I have no estimates that the government relations and energy algebras is considered the sequences of man I called France subscript superscript which again
satisfy some quadratic
relations more interest in this from students because so that the strength of this formula doesn't matter the point I want to emphasize is
that we have an algebra which is the most typical objective of linear algebra but there's a lot of slightly mysterious data and the nite for equivalence relation we should think about is something intrinsically only it's not given by you know a man from which entwined structures is given by a sequence of multimedia maps which can have all the authority many inputs so in particular it's perfectly possible that 2 different a infinity from related by an infinity equivalents which on the level of cohomology is just the identity and does nothing and in which everything interesting is hidden in these awarded so from that point of view of a basic question you want to think about is mine is infinity and the rest had of the animal in the sense equivalent to the same underlying algebra but equipped with the stroke which higher products 0 you yeah so that school is somehow about this question for exposition reasons example to comfort but I think a little bit of time settle so let we consider head home golfer has been using the Algebra I always call this case major single people are fine surface given by this equation C 3 this is for many people recognizes the pain not fiber so well he said Is it agreed capable of distinct you know what I wanted to "quotation mark I I wasn't disputing that she was correct I was wondering which OK this has a projections coming from projections of his called the seaplane and the suspended construction of Lagrangian spheres this surface due to Bolton called the matching cycle construction have a bunch of critical values of this projections places where people have said vanishes and the fiber looks like a singular ,comma X square blocks y squared equals 0 with the generic fiber is justice news chronic something that looks like the Star 1 and if I paid Paul between 2 of these critical values that avoids all the others say them then by thinking about this circle fiber which is the equator the start last 1 inside all of the fight over the interior points of gamma which collapses down at the end points to a single point then what I get is a family of circles collapsing at both ends to a point which is that to sphere which is actually a Lagrangian inside XK minus 1 to introduce something mildly more intimidating so the case Hilbert scheme for yeah as my writing from the Congress this increase what it was but why K inside the heli-skiing K points on this being the of soap schemes which all the EU projects the fraud this is bound to see still have lent can if that doesn't mean a thing if you don't worry if the heli-skiing King doesn't mean anything to you all say a word about the it present and the nice feature about this opens up that it's an apt .period variety in particularly exacts an elected manifold so the point is that I start with assault scheme here so what I want to be true is that when I look at the 1 to rule out any sort scheme which contains infinitesimal direction killed by the projections so what I actually do is inside if K was to incite killed 2 there will be a relative symmetric products which I wanna throw out in general I want to remove the image of field K minus 2 times a relatives Metropolitan him so so what is the heli-skiing if you haven't met them it's just that the singular isolation of the case symmetric Prop so you should really think and when dealing with the tape tuples of points which may not be distinct on this and finds this summer but when they come together we're remembering something about the manner in which they came together so the length of the sub is you know if they will support at different points it would just be the number of points in the substance of looking at K 2 peoples and in general and your taking the dimension of the vector space which is somehow the ring of functions almost subsidy OK across the
road or to pay .period it's just a bunch of also connect them in pairs which paralyzed joint and such the whole thing lives in a home plate said the law of the land so there's 1 on 6 points is another small to persuade yourself finally made by the Catalan each of these all I said defined the Lagrangian sup manifold in X 2 K minus 1 and if I have pay of these alongside get came branching into spheres and will the problems of those branch into spheres as a sub manifold of this complex service to the case lies far away from the diagonal I have 1 point from each of these scale sample from the diagonal but on the other hand the symmetric problems with the singular near the diagonal so such a picture defines a Lagrangian submanifolds inside this space white this is where you the research work and found a way to prepare for the most part of the structure of the program I think it was the outside some neighborhood of the big diagonal you can make it exactly the on that spend enough for the it has to be true OK lies at the final many of these crossing was matching each 1 is given the growing use of manifold and this exact sum but did not fall and the fear of Muhammad might prove a couple of years ago is that the algebra of flavor that I introduced before full characteristics so if we do offer local knowledge you have accused and so I'm not going tell you how we "quotation mark the point of the talk is of the suggest various strategies by which we might approve this awful about it which I think remain interesting open avenues so they don't realize is that this is relevant to the geometric model the linking very close to home watching not not have time to expand on that today I mean that would be an obvious conjecture that the SLM homology theories Beyond that we don't have a conjectural geometric picture of what I know not within simplistic said she will I mean there's the description "quotation mark excellence infected John base area given much Alaska which takes place inside no fiber of the adjoint quotient was notes and matrix with appropriately chosen Jordan blocks you can again Associates canonical Groningen submanifolds tobacco index by for instance the components of its Compaq caustic with a variety and the conjecture would be that the corresponding algebras full I the theorem enables us to prove that a Lincoln varying defined invite a braid group action on this space by some vital systems as a sudden flood cohomology groups is isomorphic to Kabul ophthalmology in characteristics 0 1 there's a spectral sequence anyway without just from the existence of belongings and trying some I wouldn't say that OK so so I would like to outlines of 2 alternative approaches the thinking about this 1 of which is based around the idea that they should be formal forget algebraic reasons I'm 1 of which is based around the idea that things should be formal forget geometric reasons where have proof was in some sense comparatively at home and I like it a lot because it was not of that perhaps more widely shared here 1 the previous times are debate with Mohammedi office gave a talk I accident she described bombing result is pointless nonsense for landing all let's go back to his old ones we have this nice Milner fiber before I through this monstrous held scheme you and there's no obvious collection of Lagrangian spheres in this notifiable which meet according to the source minus 1 10 section the 1st I was the 2nd so I said that I had this Blanchett's vibrations will take a critical values and that Paul between 2 critical values defined the Lagrangian to spare and the total space so I just take the king straight line called consecutively having chosen by polynomial P have of real distinct roots no I get a collection of ground units 1 L a minus 1 and so but there are plenty of the amount you decide alone ,comma Thomas that the corresponding algebra but
they proved this involves a relatively drastic way which is to say that they've proven fact that it's intrinsically meaning that any infinity structured you could fix this algebra is equivalent to the trivial 1 adjustable no interest so what if that's true that's a feature of just the algebra itself so should be amenable to somehow methods of pure algebra and indeed there is a sufficient condition for intrinsic formality formulated in terms of culture of overtures to recalled graded algebra the whole group from tank by grade and it's obtained the cohomology of a complex whose terms involves looking at multimedia maps from powers of and to a certain extent thing and program called for the job in degrees I'm to so now looks a lot like an infinity structure itself enabling the structure was built out of multimedia maps from powers of a of appropriate degree so prominent in the structure is an element of CCT With this grading this observation there on the sidling Thomas which is just just it's a nice piece of algebra that's a change so too slightly weaker Warren thanks change .period 2 minus 2 next to my ex and foundations for less than the 3 somehow all of these migrated pieces except the very smallest then it is intrinsically germany is sufficient I found wanting to prove the formalities the among Milan wants its balance both computing partial homology if you're like me 1 gets nowhere so you right to Michael who writes computer programs to compute social homology and you find that kind from unfortunately the change to call him can work this is the algebra corresponding to work in this space like a group ranking exponentially and on the other hand the remaining pieces were allowed to keep change so too ,comma zeros that's just the center of the algebra and that's so raucous grows linearly so you certainly can't prove that they're amusing intrinsic formality some and I should mention that he recently at the molecular showed that the sort of type so I the time algebras all of the characteristic commodity 2 and maybe they conjecture at the time the in characteristic of the quantity of 3 and 5 so there was this is 1 very special situations intrinsic formality seems beholden at least a bunch of other cases related to services members of congestion due to strong the grew out of conversations with men at my fans and the canonical enlargement come on course easy to describe for the natural from geometric point of view it can sitting inside Nicole had such that changed too it hand is just the center hands which is actually just a all this space Whitehead and this enlargement simply corresponds to throwing in a few more of Roger and submanifolds which need to be competitive so in Mexico minus 1 I said that there will withdrawn unions associated the poll likely so damn defines a Tuesday I could also consider just Paul which runs out to infinity along a straight line in such a signal 2 days for a copy of which was a Lagrangian submanifolds Banks can at minus 1 and so instead of looking at the original pictures I had of these configurations of pairwise destroying all you could look at configurations involving some North closed all and some of joining as a point now there exists was about to say "quotation mark perhaps crossing matching and with which will vary from being
of the best Tuesday just desks brought all to find things with the moral fewer compact Thanks and we said and had to be the direct some of local communities over these more general opening match in the 3rd column I that from the front we using small perturbations at infinity said this is still so the finite dimensional on the other meanwhile slightly .period Be True from a technical point of view defining flock of quality in the of schemes of a different order of magnitude because it now contains closed fractional come in particular you'd have to work over field north over the integers or rational so yeah I think it's probably true that 1 to take care of that the story for the closed the raunchy and and depend on the compact fine if you wanted to set up a version of this conjecture the worked in the hope the scheme that instead of introducing these known compact objects you should work in the AFC following a case of this oversee stall and look across matching so that closed in and those 2 algebras a somehow Kweisi hereditary related have become more cheerful OK so if we approve this this would be the 1st of all if this figure out rules for it would implied by an easy restriction argument that the original algebra was if this conjecture was true yeah this been around for brought before presented the stupid reasons would say that it's only deformations of the ones where you actually change the cohomology closer the supply formula Lagrangian submanifolds just physically disappear and it's some To me recently intriguing motivation of introducing more compact incinerate concrete cases where in fact there are only interested in the flood theory of the compact objects but somehow the fact that there more compact with Ron Jon's around bound up with them you some extra traction let me say that next let me say something about a potential geometric reason why sound rules form which will make contact with Jake spoken to be assessed for slightly different ways however he runs a program on a computer experiments that have confirmed the idea that it's a change to just a dimension of white and that's not true so this is not conjecture that stupid he disproved by computer experiment but is also not proved by computers because of this news about what you mentioned so what the computer can do is tell you that a change to with AEK hands has dimensions 105 5 and then maybe the 3rd 1 is 13 and after that it becomes intractable but that was all this over the years ranks the 2nd better numbers of these 1st 3 spaces the difficulty computer that's different OK so I liked as the geometric mean the formalities somewhere else in nature so if there is a cave manifold than the classical there on the Morgan and Sun said that is rational ,comma which I think so so for instance all of these years harmonic for most the Caleb are exactly the same sort of Hall of the People which employs the wage for logs of the 0 0 morning for all morning so please you just look at this she home forms of pipes stores 0 you get strictly associative algebra which somehow yeah In particular shows that the sort of P 0 of of cohomology couldn't have any nontrivial messy products so messy proxy standard instruction the formality will be defined by the failure of having a strictly associative and the argument for the Fulco would use some of the same it comes down to this sort DDE ball the says that if possible mass approximate manifold vanish coherently and that kind of coherent banishing reflecting some basic formality of the classical and from Trento knowledge so to try to think about you the cohomology with his classical infinity structure as the flower
cohomology off the 0 section in the contention pond and so you could wonder what's the essential geometric structural the contention bundle became the manifold and at least in the neighborhood of the origin of the structure carries a high structure making the 0 section of complex Groningen a whole thing Groningen in the sense of Jake's told it also has a In manifold has a skating action the contention so the fiberwise dilation is somehow at least not the only the essential feature this electrical Jake introduced complex will hold more than a variety and things at home or take 1 complex structure all I I became a family of which the corresponding homomorphic volume forms of J. Paul Simon Kay Kay in his patient Michael that get to class "quotation mark omega-3 fatty so these real audience with respect to a whole circle of forms so and that is the corresponding floor algebra full of acute tho I haven't written then the case I have in mind but I should say 1st of all this is already again interesting acts exact and these wrong the exact form fees Taylor forms we get 2 1 2 3 1 the situation because of the nature of the Lagrangian pairwise clean so what do you get somehow automatically from the because prompted this is the fact that Jake referred to no easy that if we take a generic complex structure in the circle of complex structures relevant I'm eager to and omega-3 every J. 0 because so you stick with the Hanks whose boundary lies in these ally his appeal because the boundary punctures is constant so I guess I mean every home because of a finite energy finalized area the proof of this is somehow just hope the following so let's take a look take some care for it also called for the Magpies Al-Islah grungy compatible with the complex protocols Thesiger plus sign the decay for introduced fight the minus sign that costs caused a 3 and so the point is that in this form is chosen how to treat the along with respect to life compatible complex structure Jr so if I have 1 of these following become with inside the allies while that if I pull backfired by this man then why did get to something that's built out of 2 in 2 forms of agreements this has vanished but now I just play off the fact that Michael has some positive area some conditional its evaluation the game's only goal by this Final Four vanishes the areas signal follow because the 2 times we get to evaluated on some relative philology clocks cross-ice these times with the 3 evaluated on its homology which is greater than or equal to 0 the
following evaluated and lost his -minus Ciampi finds this last and this sequel and these have to be truthful featuring the circle up you know this is infinitely many equations as I very things in life who want loans on Tuesday when omega-3 signal so my only option is actually the area is 0 so used OK so I mean you know it might be that if I chose the stupidly than you know I'm going to an omega-3 Sigma happened yeah invite you know Apple theory or I couldn't needed because like approximating by generic Peter unknown constant cuts would end up being called up when I was running through my mind at least this was obviously false near banish by varying the yeah those of the statement is that if J 0 is generic and everything is constant so I have to so fixed values under 261 omega-3 by varying the there review there will be a choice of the for I call some these equations sorry said about of the point is that has to be assessed a situation where everything interesting happens and influence theory here comes from constant matter much like the 0 section in the contention of but we know plenty of situations in which more July space is a constant maps do contribute interestingly in a situation it's by constant homomorphic clear and on the other hand the more July faces a constant this Least of these the grantee and meet pairwise clearly all themselves Kayla manifold which of so of course you know you can take to circles punch elliptic curve the only interesting homomorphic this to contribute to the Infiniti structural constant and more July space a constant disks of point and the point is also a manifolds of formal so somehow you know this data alone cannot be an by some purely general algebraic argument to imply formalities on the other hand the simplest setting that I might have this question is 1 complex Lagrangian sitting locally inside its own code tension bundle and you know the formality this is asking about is exactly that the leader of small consultant perhaps the next sentence question you could look at is when the complex raunchy and themselves of the lowest possible dimension just cut of annual working inside a hike became a surface and the situation of these Milner fibers of type AB are exactly going through the cases in which you would get so such an interesting question instead the finally just are among so they're I mean you could take the Rongen spheres in a case resurfaced for instance and also the same question I mean I think that's what you've decided that all of the curves of constant you can localize the computation to a neighborhood of the Lagrangian spheres and essentially return yourself to 1 of the In a plumbing configurations as fears of the types of these generically with some of them that there's a bit more you could do you could take costings all then the history of the world this so there'll always constant homomorphic and constant homomorphic maps can contribute interestingly to the product structure in this algebra so why shouldn't think the constant never tell you anything interesting they certainly do tell you something interesting because these things tend to be interesting algebras but the question is you know a constant method is a natural thing to have him to Sheehan's meant so small so implausible constant met the pair if 3 or more Lagrangian shouldn't be to the point so as soon as you're looking at constant lamps too high and then the structures on mightily even to the prolog you you've already put yourself in a generic situation from the point of view perturbations on the other hand if I give you complex Lacroix GM's yeah it seems very unfortunate that the top them destroying them mice geometric structure in order to achieve transverse al-Attiyah happened intersect pairwise transversely this is a situation which seems somehow Gold given for trying to approach while some sold more sophisticated technique that includes you know no 1 transverse more July spaces because here it seems to be interesting geometric structures precisely on those long friends was what was bases of constant so I'm trying to just remote that on the space Whitehead that I talked about his variety if you know what that means and that's fine if you don't then you know basically over anyway said in particular I became and as a became a manifold it rich francs to a union of finitely many complex Lagrangian submanifolds which are indexed by the crossing was matching his role I talked about it before so it contains Of course given the grantee and we shall apply so let me :colon al-Haj run indexed by the right but just as the strength of like topology question is seems
normal here falls these will Sheehan's are actually Hamilton topic the Lagrangian survival might lead by taking profits of season is notified note because yeah absolutely so so so strange emphatic apology questions this is so this the complex that Rogers adjust the Compaq although the resolution of the relevant adjoint questioned fiber of his era ' L Road Hamilton genocidal tell her that knowing that all reason to believe this is true and if this Wallstreet then no positive answer to this question in this case would give someone perhaps intuitive explanation for the formality there environment might prove that I think that the beginning and conversely you could view out there and as you know a different additional evidence that maybe there's meets to this conjecture we've proved that in some of the known review what yeah so if I reached pages 3 4 or 5 of my notes have come up after the there's a pretty observation due to pull societal and political tortured Polish student that shows that at least in the presence of certain symmetries are fixed so there's a certain in volition you can consider on MySpace of all spaces more equity vary the form of a signal to and I suspect that it probably no formal visit more too and I refer to the results of it to elect on the D and E . fighters and implied the they really do prove formality holds away from the certain characteristics and fails the holding characteristic so here as in the classical formality of chains on Micaela manifold 0 characteristic seems to be not just the multiplicity the proof Is there anything of that sort of thing you know the sort of formal directions it's recovery operations perhaps but I haven't actually about certainly you know they want you know it's for that has consequences for the global selected topology of displaced but they haven't been systematically explore that should be the weather will you could do something close to his father rational the theory I don't know it's a good idea the question you know what that was like I was very happy to have you on 1 7 maybe I mean I certainly could what is true is that and so on the fact that this algebras formal identifies it with a solve algebra of capturing his algebra which is known to be controlled by perverse chiefs from across many and by the work of breaking so you could wonder whether or not the same combinatorics you would need for the Compaq or or shows up naturally in the shooters stratification of the cross many of planes into quite UK space and if so it might follow relatively straightforwardly from how was the 1 to do that you want so I would say that I don't understand yeah trades coming from formality but the way we prove formality 1 of the mammoth skeleton made so we proved this formal by building in HH 1 clause it's properties and there is a general relationship between cultural cohomology in simplex ecology this rather geometric invariant Frederique introduced so in the simplest examples of these mummified fibers 1 can show that when viewed as an S H 1 calls the calls we build satisfies a certain natural conditions with respect to this be the operator the Frederick also talked about which is the violation condition introduced by the presence of violation has immediate consequences for instance for the existence of Lagrangian submanifolds it rules out tape I want things like that on the 1 slightly annoying thing is it's impractical GE has the restriction properties which is where these constraints come from but minutely cultural cohomology does not behave well under restrictions and somehow come variant in 1 variable in contravention of another so if you've interpreted as a safe store seems slightly less obvious to me I we don't quite have the violation condition we have something that strictly different from but 1 should somehow explore whether or not the existence of a pure vector field which is what we have just rules out the grungy ends and if you can then use a restriction technology to get the same conclusions I would guess yes but we have not I don't think that period for the yes in the entire area then there was the work here is that we have some collection run and generated by the lottery if you want and in the United States these adding 1 the still question :colon good unlike the case with this kind of job you College Station registered here on the content was not that we have 1 thing in this 1 the executives with company some sometime this in the history of irony here many of these were going to have to do so Columbia Eurojust In this in those cases this is thank you the last thing they generate the same the side effects which really generated by the sell-off corresponding compact thinks he's vanishing symbols rather than by the spheres so the long fight the objects to show up in that directed category you think of it as an allergist the whole thing with an exceptional collection just about this analysts say so you can take the dry category coherent sheets on the resolution of 0 fighter with compact support on this collection of complex Lagrangian accounts and built a break action on that essentially by hand and showed the Republic of Ophthalmology their theories by graded I would guess that starting from the fact that it's by grade you could prove formality for the corresponding algebra and assets of but they didn't prove their theory that way they took a different route 5 building pangolin variance would bypass the need for an hour more and more direct some of it from


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