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Moduli Problems in Symplectic Geometry - Discussion with Chris Wendl

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Moduli Problems in Symplectic Geometry - Discussion with Chris Wendl
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the what do you do the I wanted to mention that so I don't have actual lecture notes many costs but on the other hand most of the contents of what I'm talking about today are in a blog post that I wrote several months ago and this is the address so if you if you curious about that and in particular the 3 this is the theory that I have the right to talk about the proof now about transit solidly for Hoffa criticism but positions for generics are endangered this is a 0 sum that so that the original proof was by drag about 2004 5 2 4 I guess but there is another proof of that in a paper by were loss and appendix of 1 of his early papers but both of those to prove something last general than what I stated they don't do a stable Hamilton and structures said you contact structures and then use that assumption in essential ways sold the provide talking about is very different and the only place where it's really written down anywhere in the book so a Sony say that was the main thing that I need to prove that and the general outline of the proof it is the same as before but the main difference is that the day belongs to the space on various almost complex structures then what is the changing face of that fateful flight love wizened element in that I imagine that as many as section it's equivalent to a section of on we will have "quotation mark rather the end what bundles sides let's even a complex and he went by and about side anything that's what kind of object it is and then I would define some Flossie Absalon type of space and talk about why having some small epsilon but the point is the tangent bundle the manifold I'm working with the archives and this is a natural splitting in 2 1 complex so bundle that's generated by R and rejected fuel and other complex of model that's generated by CSI and In this forum why is entire block matrix which is 0 0 in the upper-left 0 everywhere in fact except in the lower so when I think about the universal modular space at some point so there Lee musician of the operator that he finds that space locally looks the same
as usual like that's exactly the same formula I wrote down before but why is much more constrained than it was before and why I was really what made the fear and work the 1st time around because I was able to make quite general choices of y and In contradict the possibility that there exists this section in the duel that annihilate everything the image of this so still of so well it's it's we got I mean it's the I would say it's so it's dimension is something like minus 150 compared with and the I still got rid of infinity and so so in order to actually take care of this I need to think more about splitting I have why written in his book form here but it turned out to be helpful to think in terms of the slightly different splitting which 1 can even do it completely general situations and that's to think in terms of the normal bundled home occur and now you going to ask me are you assuming that because immersed and therefore has a normal bundle I'm going signals but it has a normal bundle anyway so it is not by that Qingdao could go back to the general situation for the moment we can be mapping to any some practical Buddhism completion there isn't a whole lot of now this gives rise to an operator would have to notify the you see what's that it is clearly complex linear park Of the usual when arrives consumer operator Steve so you can take always a complex in your part of any wheeling operator and now it's not too hard to check that this 1 also is a closely type operator so since its complex linear this defines a whole lot Qandil structure on the pullback tens of resale sections hold worth it if and only if it annihilated by this complex anniversary Monday of the advantage of doing that is that we don't have already they hold sections in the picture and not as precisely this bundle but of the bundle associated to a proposition is that do you want is a section of the bottle of complex linear polymer worthless and from the same adopted you start to W and now since you started W has a whole market bundle structure and the tangent bundle romances obviously does this thing also has a whole Norfolk structure inherited and the claim is that this thing is the whole market section of the fumbles so this is a dozen observation which I 1st learned about from a peak of a mask of its institutions it's not so hard to check that this is true uh it been really just follows from the kosher immigration for you but it has a nontrivial consequences for instance this implies already if you didn't know this critical points of the day homomorphic curve have to be isolated and because simply the zeros of this section which is a whole Norfolk section with respect to Taufik structurally and was a lot more than that implies well think about what 0 sophomoric sections look like in general locally you can think of them as status and valued function on their now there's a 0 at a point that's going to be some power of the time some other non-zero Horford function and matter function has non-zero value at that point that that placing some direction it defines and complex line in Seattle so what this is doing is defining a tangent space to our curve even at a point where it's critical so there exists always is well-defined hindrances 1 1 way say that it is the there exists a smooth some bundle of said you substitute in the pullback tangent under such that team used equal to the image of the EU had
non-critical .period so the images you has a natural extension to smooth some bundled and then I can define what I mean by normal bundle it's just going to be some complex complement that's a the this and you have a lot of things that legislation Bloomberg work most of the implementation of the see you 1 of the proudest family of what the New York Times we're working on it this is the man in the house .period this is the 2nd all in so if you the hearing you're asking whether the tangent space to the family members and the whole Moffat section of month the growth this year um not quite at the detention space that section is going to be annihilated by the EU but I'm not sure about its complex on your part that's something slightly different so I I wouldn't make a claim about that but I have claimed is that for reasons similar to what you're talking about the complex in your part that will both the EU and its complex in your heart the final while they induce whole market structure on this homomorphism bundle the EU I can't bring with them here most of the time the only person so I mean I suppose in algebraic geometry you probably assume a comfort structures into In which DU is already it's on comfort part and then the answer to your question is yes this is the 1st of all so by hole-by-hole Mossad bungle II meeting the standard definition I mean local trivialization whose transition amorphous yet I had to do something slightly artificial to create that here right because the co-chairman operator have for you is not a complex linear operator because we all instructions from the destruction malts and go to work I think it actually is it is a sign of some an obvious but it's true I do think about that at some point the news the Commission is a co-chairman operators that that's something to be checked you have to check that it satisfies belied for culturing operators and then it is complex near my definition so then any complex linear co-chairman operator defines a whole Office bundle structure because there's local existence of all I think it's only going to get involved in some of the things we need to continue to construction of them well we review we're maps moved in when you did it to the host officials say all the things you want to operate the Solomons None you're talking about improving the similarity principle of something and this is this is different it is nothing his own center Everything smells the official the operator have smooth coefficients because all I did yeah take the original operator and remove its complex into a new apartment that some SMU 0 Florida so the local solutions we equation we debated during office hours at the beginning on the other I would have have to take comfort in the new part of the yet so so in that sense it the whole of extraction defining on the pullback condemnable is the 1 you think it is he only concern now is to find everywhere image DU is only defined on the complement of critical points DU is an extension of it but also to find out the .period I think that this is the point of
this discussion so it's it's due to the fact that if you have a CN valued whole Moffat map on sea there is 0 somewhere even at the point where it's 0 it has in some sense a well-defined direction right because it's it's some vanishing home function times another 1 that's not 0 that's terms direct in the last month of from the end I don't know maybe and I haven't tried to say that with the the new law this is and there are other ways to see this also that that this bundle is well-defined standards loses a little bit harder but to see that there some continuous bundle is easy if you will for instance of the chemical FY formula that tells you what you can look like in the neighborhood of critical .period they you do see very clearly that there is a well-defined tangent space at a critical point In any case the complementary bundle is what I'm calling general as normal so now that I have this splitting the EU was I'm not looking at a complex in your apartment to the original you that's a coterie Monotype operator unused RTW which is now splits into tends to normal parts so I can write is operated what form all right as the new T His attention to change part as a normal to normal part there is very the normal detention part and the other part is 0 which is claim that can easily be checked itself I claim this is what it looks like and the you Indiana and duty Indians and are consuming site operators God the bundles to you and you respectively the reason for this vanishing part 1 thing here is the cutting operator do you take sections that are intended direction to other sections in the tension direction and that's basically an expression of the fact that given a whole curve I have lots of other whole Moffat curves that I immediately signed by GE's reform frightened domain and Ching Preston encompassed all of you and I will have to this is a good time for that so take these vectorbundles complex vector bundle Thomas I have a first-order differential operator takes sections of the 2 0 1 forms value than he and it satisfies reuse yeah really knew well that evolved from what about write dealers at times people's time fatal plus the press times the data for all functions that Insee Infinity on the basis of real value not complex that's the definition of a reeling apportionment of Endesa beauty is 1 in general so the 9 these 2 new coach type operators in the picture the unity is the final line bundle dear as defined on vector bundle of rank 1 West then you start to W so that's a slight simplification so may step in the proof of zip from 3 of course we need to prove that this when you raise the bar operator this protective laws that look like with respect to the splitting now let me write as a section you W so that has a splitting of its own into attendant part normal part for up to by J T you of wine suppressed and what is that well why is acting on the domain TU is sending a from the domain into the tangent part of W and James preserving that because pro-Israel so that only it is attention part is a normal part the and since actually you commutes with Jay I can rewrite that as see you both little daylight the next time do you
I need to write that has served as the tangent import duty acting on and then there's also going to be do you then acting on it and mountain for the normal price no there's going to be more stuff In the tangent part of the target angling to ignore that stuff because it won't matter the only thing I have left is a normal part of the last term which looks something like this type and for the projection to the normal some bundle of applies to you I have checked that was a good that's what that it's behind about about because what I'm about to tell you says you don't have to care about so meanwhile following numerous surprisingly useful for a whole lot of things the operator that sends a pairwise and tension part of data to when I have options here in the top attention part to you today why tangent party you it hasn't worked that's subjective so only postpone the proof of that for the moment it is in particular what's mapping onto backing onto the space of LP Delta sections bundle of fancy linear homomorphism the signal that to you can't more the the kind of I haven't told you get cultivated exemption number usually quietly me the problem that we're dealing with is that we have a different spelling Of the tangible Moses effect Asian and the wise nontrivial in only 1 block with suspected that splitting you have to be patient I wanna know fundamentally as 1 of his operators subjective I just showing you that the top part of it is 1 of China's claimed it is what this means is that in order to show this whole thing is selective
actually Paul have to do equivalent this is rejected if and only if the map that takes
the pair normal part of Ada and why 2 this stuff that needs to be subjective which minutes mapping onto sections of class LP Delta Tau Pond Bottom the signaled that to put normal body think the simplified by task because I can well in fact I no longer have to pay any attention to tighten the space all I have the picture in the normal course the operator that's unbundled that's a little it's smaller than the original 1 plus the extra time which was very similar to a return that I already dealt with in this morning's proved so my task is simple there is also a relationship between the tangent normal splitting and the splitting of CSI versus the other stuff up there because remember 1 of the conditions in the Theron said I'd like for you to be not everywhere tangent I'm going to focus on a point where that's true Europe the blast you depend on a diet but I don't care because the residents rejected but whatever the dots whenever the dots or I can think of is the 1st time a number whenever that whenever the dots are outside the Dunnston depend on it teach they depend on on threatening with those across tournament in the splitting the slick restaurant operator at that part of the about the only condition that something was not the as someone taking notes and cited so you is not evidence such that there was a main condition no go ahead and prove that this lasting history this operator this normal cushion on operator plus the extra term is always very seductive and again why is airing an infinite dimensional space so that it no longer
seems like I've lost so much information by only having 1 block of white very because I'm also only looking at a block of similar size when I'm looking at the normal so this is not to suggest that then again there exists some objects data which is not 0 that's for orthogonal to the image of that operators and that means there will be a section of Class L Q With accentuate minus Delta of that bundle but such that do you acting on it at I had a fatal vanishes for all and that's condition was and condition too the 2nd term applied and so normal digestion of wide Tuesday a parasitic people 0 for all white only 2 conditions I have 3 conditions before 1 of which had been used the thing and have already sort of used the condition involving Tudjman space the mainland mother sitting here depends on that dispersed around here is what depends on taking space and the level wouldn't be true without that some of them back to that
little bit now again condition 1 tells me I have a weak solution of a formal adamant equation so that's going to mean fear has isolated zeros that is annihilated by some coaches operated and it was not until assumptions and very very fast so that the game is again choosing injected .period choose lie in the neighborhood of the image of that point so that we make this term .period was positive and then we get a contradiction so it can let's pick these injectors .period by which I actually mean and was being injected point for the map you about which takes my domain into them not our process I can do that because there is this level their stated saying that injected point to that matter also dance and also you but so do you and as the than another way of saying that which is convenient is this implies that you at that at the time is not intended to some of those areas that are the vindication of because by the home of record equation that's something that's true if and only if the derivatives this Matthew M. Bennett that's usually not to entry to assume at this point it's not true but also I assume you is not everywhere tended to
side so let's also assume you have the not tended to excite which means it's necessary transverse excited as the tangent space to you is complex and so excited so the chances that means ironing I have equivalent of freedom and how defined as general as normal bundled I could define it so that it actually matches CSI at this point and nearby so let's without loss of generality the normal bundle people's CSI you know when you think of the name and and that's a great .period I don't have them that's not true and the case has anything to do with the thing of it the injected printed them and the points where this is not an end to this dance so I just have to find any point where this condition is true imperturbable little bits of is also true I think that I didn't specify that I'm only perturbing and some perturbation domain I can have the city anywhere in an so and of course 1 can prove theorems that that's saying you just perturb and some particular region but then you have to be careful on precisely this detail OK so having chosen this can now take wife in a neighborhood of integers so that well the euro J is some some non-trivial mapped in the tens of directions and why is going to act on that also nontrivial weak because of the fact that so right will be problematic is if flanker attention to the other bundle but that was then why would not do anything but as long as that's not true finance those 2 spaces are not identical why acts nontrivial on something in intended direction and now I can you see is a linear algebraic glamor too make why do whatever I want and it's not production no longer does anything right why sending summer and sigh and I've arranged find the normal at the same time the last the United States itself the delineate dealing algebraic basically says there In the class of matrices that arrive In the tangent space to the space of complex structures compatible with Omega you can always so given any nontrivial vector and any other vector you can choose 1 of these matrices of maps that 1st onto to the site that's what you need to know myself so I can do this now such that was these guys match so the plant was in a product is 0 bursaries positive but 0 outside a neighborhood and then I had my contradiction against so that was a fair amount to this detail about dysfunctional analytic so reducing it to the normal bundle again this has the advantage that I reduce it to a problem involving a bundle of that has the same rank as the bundle which wise thing perturbations batteries makes it possible that I should be able to procure enough to make operator subjective I just have to be careful about the issue of whether you tangent excited not if used everywhere tangent excited than this argument can't work but as long as I can find the right where that's not true of so what about Disneyland the move I think the general case but let's consider the case where he was arrested which means that the normal bundle of you is what you think is you don't have to do anything fancy so often takes time to life and section the tangent part of you start W send it to onto this bundle so 1 of the nice things about the rest days Is that my bundle I have a very simple definition is simply the image of everywhere so actually do you know defiance of underlies the morphism and From the segment oddities of you and London and show that the landmines then equivalent to showing but the following operators seductive send them something in the tangent space to the Times-Mirror sliced plus let's say W 1 key sections of the 2nd thoughts they're going to put a subscript in Europe which are explained in a moment while at the same these other sections so invective fields of classical 1 please I think actually pretend Runnicles surface for the moment and talk about vector fields affects everyone piece which vanished at the punches anathema market .period the medicine back to L P sections of the bundle of entering a mosque the signature itself specifically sending a pair next to plus the Sigma X what's the Sigma that's the standard Koshy remand operator on the signal that comes from just the whole market structure of peace so 1 way to think
about that is it's the linear rise Costa remand operated at the identity 20 talking about hold what amounts from Sigma to itself so my claim it is not terribly deep extends this systematic staring at it is that under this isomorphism you get between the tangent bundle and piece of you the operator inflaming suggested that's equivalent to this operated here being subjected to a wide x to J. standard consumer after the accident the place where I'm really being a bit sloppy is that I change my punctured surface into a closed surface so technically I stick to stuff I should have said something about these operators acting on a punctured surface simplicity signal would be the winner is posted on operator for the identity as a map as an asymptotic Lee cylindrical map from Sigmund ought to itself I think that would have been more complicated to write down but 1 can compute without too much trouble that that map is equivalent to this map the sand 1 directed at an army of the others in the Beaufort home with the same and so to finish this up I really just need to take a closer look at the standard coach operator on the set up signal it has it's kernel and coconut all have meaning which have significant so in particular the car Of the Sigma acting on this basis sections vanish at the punches in the market .period that's just tangent space to the automorphism group at the identity and the whole was elected feels vanished those plants tangent space at the event clock signal kenyan there that's nice 1 cup but I think for a moment about what the the well the number of students who who would be the form of so yeah that's a step that escape so what's the question exactly this is the case that the majority of the incident so right now it's not case right now the signatures acting on the fields on a closed 3 months of this I put the punches back but I required attitudes events of the punches that vanishing condition is what makes this operator accruing to the 1 for the prejudices yeah the punches disappeared so the exponential rates don't make sense anymore either so what would be the space To the following action of the diffeomorphisms group of figment fixing marketplace and punctures acting on day of the audit of Jay and effective action and asking whether this tangent space well I understand call an exercise that I don't worry too much but that's the image of this offer that's just the computation 1 can do basically with the the derivatives of diffeomorphisms acting on you can get everything in the image it's operator in that way now remember the definition of stationary space it's the quotient of all the complex structures modulo this diffeomorphisms action that tells you actually there's a natural correspondence between the cool kernel of this standard operator and Tyson estates I think I called a critique my cell sites MySpace members space of all complex structures divided by the action of this group call Colonel similarly is everything divided by the image of that the cocoa and particularly after concerns I'm going to say tangent space to Tudjman which is the same thing by definition as attendance space today in Tyson flight so the right way actually to define enticements I would have been just pick any SMU family of complex structures at J is tangent to some compliments of this coterie offered standard of human that's the analytically the cleanest definition and now it actually becomes more or less a tautology that this operators suggested as you look at it what did do it takes y which is in attendance this to the title flight just multiplies it by day which is nothing take some that a few facts accented with the push operator well this JAY is still going to sit in this some space which by definition is a complement of the image of the country so that's the reason for that interjected it's just that almost by definition for this standard subsides the slice is complementary to the image of the so that Italy Saskatchewan the proof of immersed case for the monetarist case and is going to tell you it becomes more likely for this operator to be suggestive rather than let it becomes the operator of water into and what we used to think of yeah blowing up would mean what complex surreal most of the day of the week I don't want to go on and on so now I think I'm very much time left so I'm going to take a vote about what to talk about in the remaining time 1 possibility would be to talk about automatic transfer salary which was in my abstract follows onto this discussion very nicely because it's another application the normal operators another possibility would be to say that about why the exponential waits at the I'm not sure if I can do about and try to do I learned that some of the it was a "quotation mark many many say before it verbally about expansion awake while erasing key that averted racist the splitting London before every said a moment ago there was a splitting so honest and activation the tangent bundle looks like this line bundle term generated by far in the radioactive fuel plus
excited so the relevance of that is the dilemma that's fundamentals of formality preview period for instance Solomon selection of employment wildly in Chapter 2 you come across this Islamabad tells you under what conditions the linear "quotation mark monopoly on cylinder will be and that operator has to have a certain asymptotic form at infinity indicated to the so-called asymptotic operator I I think so the asymptotic operator to think about by analogy with more smaller trees that plays the role of the Hessian of Moss function functions more if and only if that is invertible operator similarly the asymptotic operator that's something that in theory is supposed to be invertible isn't only if your raid orbiters Monday generous and that's precisely the situation where the Fed home property hold but what you actually have here because but this splitting the asymptotic operator also was written in block form so you end up looking something like -minus 550 so acting on functions on the circle and some other things down here with a 0 this part is going to be convertible if and only if the that is not too generous but this time it is not inevitable no matter what it's a nice little moppet of index 0 but has and that sort of automatically there because there is there is some kind of generously built into the problem because even read from Treasury borders it as 1 direction and this additional artificial are direction is by position was 2 directions of the generously there the exponential weight is a crutch that lets us get around that because if you work in an exponentially weighted stays you're coaching operator is equivalent to that the exponentially weighted space has not quite canonical but kind of naturalized amorphous and to the usual sort of spaces but under that isomorphism because human operator changes changes precisely by plus Delta or possibly myself so for any non-zero Delta's sufficiently small this operator becomes inevitable it was the anyone else when I can I can also talk about it more in the office of sorry so that's really that's just a technical correction and actually I will also say that once applied this land and reduce everything to a problem about normal bundle now you can forget about the exponential weights they no longer play any role in the and just because reducing to the normal bundle of asymptotic Leaving normal bundle is the same size tangent directions are the same as those to you lose this part and it was left with the part that actually is invertible and now the exponential weight is still there but it doesn't matter if it's there if you just had a little bit here operator remains convertible so now you can forget about it and you still have a perfectly reasonable said home operator for the winners but so it was the other nice consequence of this normal splitting and talked about well the at the end is talking about a general whole Martha a curve in the completed some political Buddhism 1 of the things we learned is I didn't say explicitly but it is implicit in this land I stated this thing is from on regular this only in the normal course on operators rejected so now you can forget entirely about titlist license you can forget about the change in direction disrespect to the normal direction this is the operator you need to be rejected and at least 2 things crime modest it turns out that the index of this operator is the same as the dimension of monthly if people insist on quite to money actually write that down so remember do use a whole Norfolk section of a certain bundle it's not just homomorphisms from teaseller . 2 new star W but actually it has its image by definition in the tandem part I'd encountered sections let me make my life easier and remove the docks that's just talk about a close to an answer for the moment and I can compute C 1 of this bundle that's going to be the number of critical points the algebraic number of critical points each of them counts positively possibly with some extra positive order and that's the 1 that is 1 which see 1 of 2 yuan dynasty 1 of 2 said that's the only thing of so the difference between C 1 of you anywhere characteristic is exactly this number of critical points and that means the generalized normal Model S C 1 equal to planned usually call 1 of you that's by definition C 1 of the pullback tangent bundle -minus see 1 of the tangent part which now is for the characteristic class ,comma critical points but see 1 minus directed client number could play so having more critical points make C 1 of my normal bumble smaller now .period is I can say a lot more see if if the normal bundle happens to be a line bundle so question
if he is aligned bundle when must of course renowned operator on the edits well why should you believe there is any such conditions line bundle the index formula involves the 1 which is also related to some count of zeros sections restrained use that fact indeed across the operated call that is going to be subjective if and only if its formal and injected quality stock so was a former larger I don't really need to know much about this but just injured generalities the formal adds it's something that conjugate to because the remind type operator on some vector bundle the indexes might be in so this is because the agreement I operated on some bundles let's call it what you had such that the index increased thought it was minus the index and this is a very unusual sort of many costs for me because I haven't until this point needed to claimed that I know what the in the borrower's but now I really needed that we know the index these really nasty 1 you continues to monitor the hatch from all this the index of the Stock by remind rock talking Metaline model so it's going to be there for algebra geometry as this is the real index not complex and that's where the characteristic of domain plots to see 1 of the hat that's minus index of the originality which means minus characteristic of the domain plus to the 1 of the original model and this tells me that to see 1 the had equals -minus Florida characteristic tends to minus you 1 what's erased the choose just say this is minus characteristic minus the 1 of these now a loser if she won the heart is negative then the star necessarily injected twice that it's a counter zeros and so if there's something in the kernel of the star that's a whole more thick section for some definition of whole it's 0 is necessarily positive so it cannot have any since well we can't exist in fact because of the negative counter zeros would not be possible this is where what I'm saying is absolutely unique to a complex line bundle you can't do that so that of course is equivalent to saying original operator said so I get this criterion that says that if he wanted he had negative now was name that means this already characteristic plus C 1 of the it is possible that implies the suggestive now let's put that back into what I had in the nonlinear setting let's say suppose the dimension of my family and syntactic manifold is floor so that my normal bundle is a line bundle and the criteria now says C 1 of the generalized normal bundle has to satisfy relations "quotation mark characteristic plus see 1 of the normal bundle and got a formula for it up there somewhere that says it's the 1 of you -minus where characteristic minus number of critical points so for characteristic gets canceled out I have see 1 of you and I have a number of critical points this is what has to be positive if this is positive then it sends you must be and that's completely general my and there is a version of this proof for Mercer what occurs the paper by Whole Foods Osika arrived and he was already and we sketched by Abramoff and his 1985 paper this version that allows the critical points appeared 1st in a paper by a vast regions of Titian and then there is also a punctured version written on paper of mine but I think I don't have time to talk about defeat asking questions about the only other thing I want to say so this does it does not make any assumptions all about you being somewhere this sometimes can be made to work for mostly covered homomorphic curves and let me just mention 1 case where there's actually useful so so even rewrite this formula a little bit because I have the
index of the equals so in this four-dimensional Casey dynasty characteristic
Plaza 2 1 so this criterion is equivalent to this number being greater than minus the other characteristics plus twice the number critical points so let's right that to the to test flight number critical points that's a sufficient condition for Francis Audi now punctured version says the following the only difference in this whole discussion with the punctured version is I need to work a little bit harder to get my criteria non-aligned bundle for what guarantees across have operators being rejected and I talked about in terms of counting 0 of full Norfolk section now see 1 of line bundle over punctured services 0 it's it's a trivial bundles so that somehow not the right question to ask but these whole market sections also satisfy asymptotic conditions that have to do with the asymptotic operated determined by the rape audits so there winding numbers associated to those rate there are somehow predetermined and those related to ,comma tendencies those applied kind of constraints you need so what you end up with you explore all of that is a criterion that says if the index is strictly greater than twice genus minus 2 plus the number of punches where the calmly saying the index is leaving plus twice
the number of critical points 10 years later my last comment for today's going to be following exercise the people of the world the food that for generic J. in four-dimensional simpatico Buddhism all index 1 day cylinders are regular as part of the exercise kind but the problem with this term accounts a number of critical point it's not really something you have any topological control over however if you looking at curves of low index there are more things you can say because being immersed is a generic condition sense also something I'd be happy to talk about in office hours if anyone wants to talk about their results that say looking at someone injective curves the the sub space with the space somewhere active curves that are not numerous we inside manifold a certain core dimensions "quotation mark mention at least 2 compared with everything else so if you idiot index you can exclude curves of critical point entirely and then you have a much better chance of being able to apply this criterion so and other places where you can look at the answers to the exercises Joseph thesis because she made considerable use of this sort of recent accusations she has made use of this because it means you can honestly count index 1 home of its is even if they're not because they the better written on it we forgive there were also the last thing I want to over the index of the unconfessed problem yet so we literally the virtual dimension of the model is based upon the evidence that's that's what I mean by so bad so in immersed case that happens to be identical To the Fred home index of the normal course on operator and that's that's something that's easy to but so am I and what