 Thanks for the introduction. I'd like to thank the organizers for allowing me to talk. OK, so a lot of this will be joined with Michael. And it was helpful. He came with his obstruction, bundle gluing, and intersection theory from ECH. He was kind enough to lend them to the contact tomology. All right, so we're going to start off with a contact manifold. For most of the talk, it'll be three-dimensional. But in some parts, I might get to tell you about what we can do in higher dimensions. This is going to be non-degenerate contact. And it's going to be co-oriented, meaning that there's a global one form that defines the contact structure. And for grading reasons, let's pretend or assume that the first turn class of the contact distribution vanishes. And then R is going to be my rave vector field. OK, and then I'm going to have J is an almost complex structure on R cross M. And I'll take d e to the tau lambda. So here, tau is going to be my coordinate on r. It's going to be translation invariant. Yeah, translation invariant. So I have d tau R is going to be J d d tau is going to be equal to R. And I want J restricted to, I'll just say J is d e to the tau lambda compatible. OK, then we want to count cylindrical curves. So crystallized asymptotically cylindrical curves, R curves, the domain is just going to be a cylinder. So it's not going to be super exciting, but they're still fun to look at. So I'll denote the space of let gamma plus gamma minus be periodic rave orbits. Mj gamma plus gamma minus is going to be the set of J holomorphic curves from a cylinder. R cross S1 J0 into the simpletization equipped with this d lambda R invariant, almost complex structure. And then we want du of J to be J of du. So we want them to be pseudo-holomorphic. And then we want the limit. So if we take R to have the T coordinate, S1 to have the S coordinate, we want the limit. As S goes to plus or minus infinity of the projection of U on the contact manifold component to be a reparameterization of the rave orbit at gamma plus. If we go to positive infinity and gamma minus, if we go to minus infinity. And then we want the limit as S goes to plus or minus infinity of the projection to the R component of U to be plus or minus infinity. So you want to think like if my contact manifold was something stupid like a circle, then you're counting curves like this, gamma plus gamma minus. And then there's also going to be an R action that technically you have to mod out by coming from translations on the target. And so when we go to look at the differential, there will be this translation action. But a lot of times I'll ignore it when it comes to talking about index calculations and stuff. OK. So anything other than the gradient dependent on your assumption that C1 equals 0? No. So I can just drop that and be happy with the Z2 gradient? Yeah, so you can drop that and use the Z2 gradient. Of course, when I define a dynamic convex contact form, you'll need some sort of way of making sense of the gradient on contractable orbits. So the gradient is given by the Conly-Zehnder index. On our chain complex, we have that the virtual dimension of this moduli space is given by gamma plus minus the gradient of gamma minus. And here the gradient is an SFT gradient. So this is going to be Cz of gamma plus n minus 3. So if the dimension of our contact manifold is 3, then you get this is just Cz of gamma minus 1. So then I can tell you what the chain complex is. So this is going to be generated by only good rape orbits. So we take the set of all rape orbits, and then we throw out bad rape orbits because they're bad. And so to define a bad rape orbit, it's a little bit annoying. But so the Cz index of a rape orbit can either have the same parity under iteration of your orbit, or it can flip flop between being even and odd. I'm sorry, odd and even. If it flip flops, then you throw out the even multiple covers. So bad rape orbits covers. If you want the real definition, or you want an example. So in dimension 3, bad rape orbits are even multiple covers of negative hyperbolic guys, meaning that they have the eigenvalue of the linearized return map. If we restrict it to the contact distribution, is negative and real. Is there a reason for your v there? For which? The ccvq, like in your top line. Is there a reason? It's not a v. It's not a v. This is the vector space generated by all the rape orbits, and you throw out all the bad rape orbits. Yeah, sorry. He just thinks the star is a v. Never mind. It's clear enough. Is there another way for why are bad rape orbits bad? So if you keep them in the chain complex, even if you had all the transversality in the world, you'd never be able to prove invariance. But secretly, you can't orient curves that limit on bad rape orbits unless you have some sort of a parametrization you're using of your bad rape orbit. And so when you try to assign orientations, you can get both positive and negative orientations assigned to the same cylinder, and this causes a headache. And in Orbithold Morse theory, you can kind of think of the bad rape orbits as being the bad points whose gradient flow lines get reversed underneath whatever action you're modding out by. Any other questions? This motorized space is a factor between more often parametrization of the cylinder, right? Yeah, so that is kind of incorporated by the fact that this is the projection onto the contact manifold as a reparametrization of gamma plus minus. So that already includes that. And then the differential. So there's actually two equivalent ways over q of defining the contact differential. So either we can define it to, so let me just write this down, and then I'll say, OK, so either you can define it by looking at the multiplicity of the top rape orbit, or you can define it by encoding the multiplicity of the bottom rape orbit. So and this epsilon u is just supposed to be some choice of coherent orientations. M of u is the degree of the covering map of the curve if your curve u is not somewhere injective. And the multiplicity of your rape orbit just means you've had to iterate some simple orbit some number of times, and the multiplicity will be the number of times you've iterated a simple rape orbit to get to x. And so it might look like these coefficients are defined are not integral, but they are in fact integral. And so you can see that, because if you have a k to one-fold cover, where you have x is gamma, let's see, plus to the kp, and y is gamma minus to the kp. This would be a k to one-fold cover of a cylinder, gamma plus to the p, gamma minus to the q. And so the multiplicity of x is kp. The multiplicity of y is kp. The multiplicity of this curve u is k. And k obviously divides both kp and kq. I'm just ready to guess that m of x equals 1, if x is a simple rape orbit. So maybe simple hasn't been defined, but that just means that the map x, which is r mod tz, into your contact manifold is injective, and m of u equals 1 if u is somewhere injective. So that looks like beautiful theory, but it's not really so beautiful. So some of the issues that are going to arise is that we're working with multiple covers. Are there a question? So can you explain the difference between the two? Well, one encodes the multiplicity of the top rape orbit, one encodes the multiplicity of the bottom rape orbit of your cylinder. And it turns out that they're equivalent, sort of if you had all the transfer salinity in the world, that would mean that your differentials were defined and you actually got a homology. And then on the chain level, you could just divide and multiply out by multiple seed of rape orbits to pass from one chain complex to the other chain complex. So the homologies are isometric under this? If you live in magical transfer salinity. They can be of two different branches. Yes, but they're equivalent. So why is this theory possibly not so correct? OK, so the first issue is that we're working with multiply-covered curves. And Chris spent many lectures telling us, as well as Dusa and others telling us about the horrors of multiply-covered curves. And then also, I think Michael mentioned this a few times, but you can look at multiply-covered curves and they can end up having negative index, even though they live in, even though they have to be there and you can't just use a generic perturbation of J to get rid of them. And so this means that compactness issues are going to suck. Well, maybe they don't suck. Maybe they're beautiful, but they make the theory harder. All right, so everyone's favorite congerum from 2000, which is in a paper by Ali Ashbur, Gimental and Hofer, says that, assume there are no contractable rape orbits of index minus 1, 0, or 1. So this is this grading. That's the Conley-Zander index plus n minus 3. And, OK, it also started by saying, what, y, c, v, a non-degenerate contact manifold done. The differential is well-defined. Of course, no one really said which differential they were using, which adds another layer of confusion, but that's OK. J is generic, that you actually get a homology. So dq squared is 0. And this homology, which depended on your contact form, your choice of J, was actually an invariant of the contact distribution. So EG independent of the choice of lambda and almost complex structure J. I think should be fair. They didn't quite say that, because they said that some abstract perturbations of the modulate spaces would be needed, but they wouldn't go into the details of that on that paper. Because this one is too much of a problem. Well, it doesn't even appear in this form in their paper. It appears as a series of lemmas or propositions. They said at the beginning, this is all soothing some perturbations of the name. Well, at the beginning it said that this is all conjectural. So this is why it's a congerum. So it's a conjecture of A plus. So the conjecture is that this statement can be corrected to the other side. Well, the other problem is that some people then said you don't need abstract perturbations, and this works anyways. Is anyone happy? No? No one's happy? OK. So what is this beautiful compactness that I alluded to? So this is Bourgeois, Lyoszberg, Hofer, Vizatsky, and Zander. This is a result from 2003. Again, I'll paraphrase this result. So there's no bubbling like in Hamiltonian Fleur theory, but we do have breaking of curves into buildings. So Michael drew some beautiful buildings of asymptotically cylindrical pseudo-holomorphic curves. And so the heuristic reason for why we have this breaking is because there is a maximum principle in the R component of the simplectization, but you can grow minimums. And so if you try to compactify the space, you're going to be allowed to develop new punctures at the bottom end. So you had some index two cylinder. You could conceivably start to have the picture people draw. So you have some sort of a minimum developing in the R direction. And so then if you want to compactify the space, you could pop off a plane. And this would be allowable elements of your compactivification if you could not sort of exclude it for other reasons. And then this makes us sad, broken cylinders in your compactified, modulated space of index two curves. I have a dumb question. Why is this breaking and not bubbling? How much? Because you have cylindrical end, which is, as I said, has a real boiling parameter instead of a complex one. It breaks like a trajectory of more stereo or core theory and not like a bubble and a grimoire. OK. That's really great. So I'm not going to draw a complicated sequence of buildings, but you can imagine that sort of, well, Michael was talking about how you can have branched covers of trivial cylinders. Those guys can have index zero. So you could be in some situation where x is gamma squared, z is gamma. And then the cylinder could break to be gamma squared, gamma, gamma, gamma. And this could be a holomorphic plane of index two. This guy has index zero and this guy has index zero. So this would be an allowable compactification. And in dimension three, there is really nice iteration formulas for the Conley-Zander index plus automatic transversality. So I'll explain in a minute how you can salvage this theory in dimension three. But in higher dimensions, you can have cylinders of index one and you take their multiple cover and they suddenly have negative index. And then that means that you, in other elements of your compactified modular space, are going to have to include things like a chain of cylinders where you just have to end up being able to add to two at the end of the day. So that makes things complicated. And philosophy with a lot of this stuff is, well, if we only had somewhere injective curves, you just pick j generically. So all of my modulized spaces have to have positive virtual dimension because we're working in a simplectization and have this r-action. And the only way that you can add to two with positive numbers is one plus one. And this original assumption about no contractable orbits of index minus one zero one come because this sort of building is not excluded by the assumptions, but you could have some building where you have one, one, and one. So this is like why the assumptions in the congerum exist. And then the zero and minus one sort of come up when you try to prove invariance and homotopy depending on what level of transversality you want to assume. So it turns out that in dimension three you can get automatic transversality and index calculations to work favorably. And so in my thesis, I found not a very large class of contact forms, but a class of some contact forms where you could actually define contact tomology. So sort of the definition I came up with is we say a contact form is dynamically separated, provided the conlesander index of gamma is between three and five for gamma simple and contractable. And every time you iterate gamma you're just increasing the conlesander index by four. So this holds and then plus you can do this if you, so I guess I should say if gamma is non-contractable have analogous definition but you have to keep track of free homotopy classes of rape orbits and it just becomes a mess to write down so I'm not going to. Is there a question in the front? And then the theorem that I proved is that if, so this condition, this is usually true up to large action. And action is the integral along gamma of some rape orbit. So by up to large action I mean for all rape orbits of action less than or equal to something you have this dynamically separated property and these guys all for the most part are rises. There's a natural class of perturbations you can put on pre quantization spaces. So this is where these guys come from. So if my three lambda is non-degenerate dynamically separated then and J generic then the cylindrical contact homology is defined and sometimes up to like some sort of. So if you don't have this dynamically separated for all of your rape orbits but only up to some large action then you can only define cylindrical contact homology up to that large action and index and then there's a way you can take direct limits to actually sort of compute it. Invariance under the choice of J and dynamically separated contact form. Okay, so this is not so awesome because it excludes a lot of examples like dynamically like certain dynamically convex things like ellipsoids. So you team up with someone that knows more than you and then you prove a theorem. Hook thing is annoying. Okay, so a dynamically convex contact manifold is something that Hoper-Vazatsky and Zander studied in the 90s and what Katrin knew as HWZ papers was when she was a graduate student. So a dynamically, so Y3 lambda is dynamically convex provided the first churn class of the contact distribution restricted to pi 2 of Y is zero and the Conley-Zander index of gamma is greater than or equal to 3 for all contractable rape orbits. Okay, and so an example is any strictly convex hypersurface in R4 with the standard symplectic structure which is transverse to the radial vector field. So this is a nice class of contact manifold because I erased the theorem but it's basically without sort of a theory of abstract perturbations that's going to be the class of contact manifolds so that all the contractable rape orbits give rise to planes that all have to have virtual dimension at least two. So this would be sort of like a nice class of manifold you'd like to prove contact tomology is defined for an invariance in dimension three. Okay, so, oh yes, okay. No contractable, condition A, C1 of pi 2 is zero or B. So what Michael and I proved is that if Y lambda is dynamically convex, J is generic, then the only buildings of index 2 in R cross M are either just a cylinder, a once broken cylinder or if you, okay, I should also write this. So we're going to assume that Connelly-Zehnder index of gamma is strictly greater than 3 for gamma non-simple and contractable and I'll explain why in just a second or you have a pair of pants, a plane, and a trivial cylinder. So gamma squared gamma is zero, zero, two. Okay, so we have this extra assumption that there should be gamma D, D minus 1, sorry. Okay, so we have this assumption that the Connelly-Zehnder index is strictly greater than 3 for gamma non-simple and contractable. We expect that it's removable, but we have to do a little bit more estimating things to make sure that something bad doesn't happen in the next blackboard, but sort of what we proved is that the only buildings of index 2 are either an unbroken cylinder, a once broken cylinder, or this pair of pants configuration. If you assume, if you don't have this assumption, then your pair of pants configuration can have like this gamma could be gamma to the D1, this could be gamma to the D2 and then this is just a D1 plus D2-fold cover of a trivial cylinder. And the second part of our lemma is that no index 2 cylinder can limit to the third configuration. So we didn't prove that there are no buildings of this type. There certainly are, but we just showed that there's no index 2 cylinder that could limit to it, which means that as a corollary, the cylindrical contact tomology differential is well-defined and it squares to zero. And this is in a simpletization, so the reason why we don't have invariance yet is because this sort of analogous lemma in cubortisms for sure does not hold. Any questions? So a priori, what are the possible configurations for that third one? I mean, I'm looking at this and you're saying if it can't be the case that a cylinder, if it can't be the case that a cylinder of index 2 limits on the third configuration, I'm looking at that thinking, okay, well, if in some world I could perturb and actually glue that thing together in some sense, then it would be an index 2 cylinder. So therefore, if you, like, what kind of perturbing do you want to do? Well, that's sort of my point. So if you were doing obstruction bundle gluing and you decided to try to see if you could do this, the number of ways you could glue it together would be zero. Actually, just to, that's not how you prove it, right? No, no. It's approved by contradiction. Sorry, so we use, yeah, you use intersection theory. You say, like, suppose you have this curve living into this, then you look at some intersection theory properties and prove that you get a contradiction. Therefore, your index 2 cylinder can't limit to a building of that type. True proofs are actually the same if you look deeply enough. Yeah, and since obstruction bundle gluing goes back to winding numbers and, actually, I have, well, can't the obstruction bundle gluing, can it only tell you that the sign counter phrase to glue is zero, whereas this is really telling you there's no way to glue it? I think in this case, you'll actually see if the obstruction section over the adhesives could go to that too. And where do you use the intersection theory stuff? Oh, the intersection theory stuff is to prove that no index 2 cylinder can limit to the third configuration. And where is that to use the fact that it's dynamically convex? Oh, so we use the fact that it's dynamically convex to prove that the only buildings of index 2 are of those three types. And then you also, I mean, yeah. And then you're also using automatic transversality and some index stuff, improving that those are the only three types of buildings. And that's also what's getting you the transversality you need to say that the differential is well-defined. So we also prove that all index 1 and index 2 cylinders and simpletizations are cut out transversely. And do you need dynamic convex for that? So you don't need dynamically convex for that, but you're going to have a problem with looking at your compactified, modulized space of index 2 cylinders. Like, then I'm going to have way more things, and I'm not going to be able to prove that d squared is 0. Let's say you can know that you only have these three kinds. You can only get that if you're dynamically convex. We all have bad vision of people in the front. But what are the indices of the curves in the third building? Zero, zero, and two? Yeah. I didn't want to put the zeros here and here, because then the people in the front row would ask if those curves had genus, and I was counting cylinders. Yeah, no, it was good. I just need plastic. The top bubble there is a branch cover of a trivial cylinder. Yeah, I should have. OK, that's fine. OK. OK. He's a trivial, trivial cylinder. Everyone happy? If you're not happy, I'll get on my hook. I actually really like holding the hook, but I think that makes for a slightly scarier talk. Oh yeah, and I guess, so I was going to move on to the non-equivariant part of the talk, but do people have any other questions about what goes into the Waman corollary? I guess I should maybe upgrade it to a theorem, but anyways. OK, so two, we're stuck on invariants, so we need to do something. So we'll try to define, so we'll define stuff non-equivariant. We'll define a non-equivariant version. Oh yeah. So to clarify, this normal approach looks bad because you can't prove this sort of lemma in co-bordisms. You also don't have the sort of automatic, I mean you have some automatic transversality in co-bordisms, but it's not good enough, and you're not going to be able to prove the chain homotopy equation, so invariance is not really going to go so well. So what we can do is we can define a non-equivariant version and when we set up a non-equivariant version, it's kind of analogous in Hamiltonian Fleur theory to when you pick a time-dependent Hamiltonian, so all of your curves end up being somewhere injective, so we're going to do something similar in the contact world. There's still going to be some things we have to work out, but they're doable, and then it turns out that there's a way to take an S1-equivariant version of this non-equivariant picture, prove that it's isomorphic to classical cylindrical contact homology, and in a very roundabout way, get invariance for the classical cylindrical contact homology. Okay, so this non-equivariant theory is kind of look like the positive part of symplectic homology, whereas this cylindrical theory, if it was defined, it would look like the S1-equivariant version of the positive part of symplectic homology. But we don't want to use symplectic homology because we'd like to define something directly in terms of the rape data and in terms of asymptotically cylindrical pseudoholomorphic curves as opposed to looking at the Fleur equation. Okay, so we're going to take J here. Instead of using one J sort of for all time, we're going to take a domain-dependent J. So J, T for T and S1 is going to be an almost complex structure. And we still want for each time T that it goes to the rape vector field, and we want for each time T that it's D e to the tau lambda compatible. And then we're going to be counting, well, we're not going to be quite counting pseudoholomorphic curves where I just throw this T dependence in because that's not going to, you know, automatically get us a non-equivariant version. We're going to have to be a little bit more careful than that, but the moduli space we're going to start to want to look at and put point constraints on is going to be this guy where we're using a domain-dependent family of almost complex structures. So you just put a T in your J-homomorphic curve equation. Okay, and this T depends, oh shit, sorry. My R coordinate should have been S and my S1 coordinate should have been T. Now it makes a lot more sense. Okay. Maybe this is stupid, but what happens if you make lambda time dependent? Well, so this is forcing in some sense, you can't make your lambda time dependent. This is like the closest you can get to forcing your periodic orbits to be time dependent. I can't make lambda time dependent? It's a contact, like how would you make it time dependent? It's just a contact form. Like it could take a time, a group of contact forms, give them the same context, right? Well, I would rather take a loop. We're not looking at period one orbits, we're looking at arbitrary period one orbits. Oh, yeah, that's the other thing, sorry. I mean, the point is now all the curves that we're multiply covered are no longer going to be multiply covered for a generic family of almost complex structures that depends on the domain. So the good news for a generic family, j gamma plus, gamma minus is a manifold of dimension equal to the Conley's Ender Index of Gamma Plus minus the Conley's Ender Index of Gamma Minus. And I broke some symmetry, so it's got an extra dimension on it. Right, and so you're going to be working in a situation where the, so there's a paper by Fleur, Hofer and Zalaman that gives you transversality in the Hamiltonian Fleur case, and so this is sort of the, you can't pick a time dependent Hamiltonian, but you can pick a time dependent almost complex structure in your J-homomorphic curve equation, and so that's how. So all your curves are going to be somewhere injective and ta-da. Okay, but you're not going to be able to define the differential sort of in this naive way here. So let me tell you how to define the differential. Okay, so to define a differential, we're going to need to throw in some point constraints, and this is kind of how we're forcing a parameterization of our ray orbits in terms of time. So, okay, so here's my curve. I'm going to be able to look at u of r cross zero, and this is going to pick out some line on my cylinder. Okay, and you can use this to define evaluation maps from our modulized space into the image of gamma plus minus, which we define to be the limit as s goes to plus or minus infinity of the projection of the curve when we look at time zero. Okay, so this is supposed to be like when t equals zero, and this is some point that's in my raybar met gamma plus, and you can do the usual thing where you can map this evaluation, map the sense to an evaluation map after you quotient out by r, because even if we do define a differential, we're still going to have to mod out by r where this r is coming from the translation in the simpletization direction. Okay, and then my chain complex. I no longer have to throw out all these bad ray orbits, but I am going to have to have twice the number of ray orbits in it as before. So, you can kind of think of this as a more spot version where your Morsebot manifolds are your raybar bits, and you put a height function on s1, and so that gives you two critical points. One critical point has index zero, one has index one, so we're just going to kind of... That's not really what we're doing, but it's the vestigial motivation for defining our non-equivariant chain complex to be generated, so it has two generators associated to each rayb orbit. So, gamma check and gamma hat. Gamma is any rayb orbit, and then I'm just going to arbitrarily shift the grating up on my gamma hats by one, so we have that gamma... The image of gamma... Okay, I'm just going to say that gamma... Conley-Zehnder index of gamma check is equal to gamma. Conley-Zehnder index of gamma hat is equal to the grating... I reversed this. So, I should have written the inequalities the other way, but the grating on gamma... I'm getting confused with hats and checks. Okay, so the grating of gamma hat is going to be equal to the Conley-Zehnder... The grating of gamma check is going to be equal to the Conley-Zehnder index. The grating of gamma hat is just going to be the Conley-Zehnder index shifted up by one, so the Conley-Zehnder index of gamma hat and gamma check doesn't make sense because it's the same underlying rayb orbit gamma. So this should be the correct formula. So you just think that I have gamma check and gamma hat are formal variables associated to the rayb orbit gamma. Okay, uh-huh. Yeah, this is any rayb orbits. You allow also the bad rayb orbit. So it's just like in symplectic homology you would also see the bad rayb orbits. Okay, and this is kind of... So this non-equivariant picture is sort of not original, but it, you know, is an adaptation from stuff by Bourgeois, Lyoszberg and Ekholm, and also Bourgeois and Wantscha. Okay, all right. And now my differential is going to have a block decomposition and I'm going to have sort of stuff that goes from a check rayb orbits to check rayb orbits. So there's four different ways. Sort of you can go from... Let me just write this out and then we can talk about it. So hat, this is just Z generated by this guy. So I promise that you'll see this evaluation map and I just wrote it down and show up. So it has like a triumph that you no longer have to throw up with bad rayb orbits? Is that something desired or is it just a quarter of this? I mean, so it is desirable and it turns out that this S1-equivariant version, like once we figure out what the S1-equivariant version of this non-equivariant theory is, it's actually defined over Z. And so it's going to have more information coming from bad rayb orbits because it's defined over Z. And it turns out that if you tensor that theory with Q you get the usual cylindrical contact tomology theory. So it's a quirk, but I guess it's good because you get more information. Okay, so what do each of these terms count? So d check, alpha check, beta check. So here you would want the degree between the degree and index between alpha and beta to be one. So you want to count cylinders where you are going to, right, so you want to, you're going to fix a point on the underlying simple rayb orbit associated to alpha. So this is some point on the image of alpha bar is a simple rayb orbit to alpha. And then we want, so this is going to be curves U in this moduli space going from alpha to beta mod R with the positive evaluation map of your curve U being equal to p alpha bar. And we're also going to have to count Morse bot buildings with a cyclic ordering condition. So the evaluation map constraint is supposed to be this sort of point where p alpha bar needs to match up with pi y of U of s zero. And then we'll count these guys. And then we'll also have some buildings that we have to count where you've got a cyclic ordering condition, meaning that you want f minus of the top curve, f plus of the bottom curve, and whatever point you're picking on this intermediate curve. Let's call this guy gamma to be ordered this way as you go around to gamma. So you're going to need to orient to your, like a direction for how you're going around your rib orbit. And then d check of alpha check beta check is going to be something analogous, except instead of having the positive evaluation constraint, you're going to have the negative evaluation constraint. And then d plus alpha check beta hat, this has no constraint. So this looks like the two bad map that you see in the Birch-Wile-Yoschberg-Akholm formulation. So if we look at d plus of alpha check, alpha, sorry, alpha hat, alpha check, you get two points if alpha is bad and zero if alpha is good. And d minus alpha check alpha hat, this has constraints at the top and bottom. And this non-equivariant theory is not really so easy to count because not only are you sort of counting cylinders that either have one point constraint, two point constraints, or zero point constraints, you also need to count more spot buildings that have the cyclic ordering condition. And these more spot buildings, they're not going to be of infinite length because you only have a finite number of rib orbits in each Conley-Zehnder index, but they are going to be some sequence of buildings that you have to figure out how to count. But it turns out that, so if, let's say, okay, so theorem is that, oh, and I should have, sorry, this d minus here, so this also in addition to counting u and m alpha beta with f plus and f minus constraints and more spot buildings, it also will count ways to glue an index 3 curve using obstruction bundle gluing that looks like a branch cover of a cylinder. This is zero, two. What is the j on the cap there? The, it's a 2. No, it's a j. So you fix a family j of t on the remains. Yeah. So it's a bubbling off at a point. Yeah. Yeah. So some j would bump it off. Yeah. Is it clear why you can only have one plane as well? Yeah, index calculations. Is the reason that you've defined this chain problem is to ever see that the high dependence means you can do an isotropy? Yeah. So your curves are always going to be somewhere injective so you don't have the isotropy anymore to deal with. So for y3 lambda dynamically convex or y2n minus 1 lambda, oh, I don't want to say that, this non-equivariant contact tomology in j generic is a chain complex and the homology of it is independent. This is a family of j's of your choice of family of j's and the dynamically convex lambda. Let me tell you how to relate this back to a cylindrical contact tomology in a way that will make everyone much happier than they are now. So if there is sufficient automatic transversality, then as needed to define the classical cylindrical contact tomology differential, then we can use for some generic choice of j, we can use this j to look at this non-equivariant contact tomology. And what's nice is that it turns out, so in this case, d check looks like the standard cylindrical contact tomology differential and d hat looks like the other cylindrical contact tomology differential and you end up having opposite orientation, but that's not so important. But it turns out that you can show that so if y lambda is not degenerate, dynamically convex j generic, then for d0 equal to dq minus d, well, I did not use good notation for this. So the first differential, this is supposed to be the second differential that I told you, d plus d minus and d1 equal to 00 multiplicity of gamma 0. I'll explain what that is in just a sec. You can actually get a chain complex by taking the non-equivariant chain complex, tensoring it with the formal power series of variable 2 that has degree 2. So here u is a formal variable of degree 2 with dz being equal to d0 plus d1 and this is j. This is a chain complex. So then your differentials, they never saw bad orbits before. So they ignored them in the second one. So these ones. So this was kind of a lie. So you still will be able to see your bad orbits and you're not going to have the problems that you had before with orienting them because you have point constraints running around that are going to allow you to deal with orientations. So I'm doing a bad job explaining this because there are a lot of details I'm omitting. Yeah, but then, so the degrees of d0 are coming from a complex where the bad orbits. Okay, this shouldn't be, I should have just written, this isn't technically this differential. It should be, I should have written something like this is counting, my notation is really awful. Yeah, we'll just use d hat and d check and they, but they have a more familiar expression in terms of multiple cities of ray orbits and multiple cities of your pseudo holomorphic curves. So what I should have written here is that if j is, if we can replace j with a single j, then d check counts in mj alpha beta, you still are going to have your checks and hats with a point constraint at the top. And you'll get that d check of alpha check is going to be the sum over the multiplicity of alpha divided by the multiplicity of your curve U that connects them and now my font has become microscopic beta with the orientation. So the chain complex is still this thing that we're using and then the differentials just have more familiar expressions in terms of the previous differential, but that was a very unclear, unhelpful way that I explained that. Can I say one thing? I mean the d hat and d check, they're the same as the differentials before when you're going between good orbits and then something else. Yeah, and you can show that when they do encounter bad ray orbits nothing horrible happens. I should finish the statement of the theorem. And the homology z of this chain complex is independent of lambda and j and a proposition is that CHz tensored with q is isomorphic to CHq. And so this is how you end up getting invariance of cylindrical contact homology and I will end. I mean if you don't have something that's dynamically convex then you're going to have to be working with like full SFT or symplectic homology. Yeah, so if you take the connect sum of two tight spheres you're no longer dynamically convex. So this sort of result doesn't work for connect sums but I mean cylindrical contact homology was never built to being invariance of things that weren't dynamically convex in dimension three and so there was a result that Ko mentioned in his talk where he said that if you have this assumption about conly zander indices then the positive part of symplectic homology is an invariant of the contact structure that's exactly this dynamical convexity condition and it has an analog in higher dimensions. What's your favorite manifold way to compute it? The sphere. You can also compute it for like lens spaces and like pre-quantization spaces. Oh, so well I guess I can just say it in words. For the three sphere you have Q in every even dimension that's greater than or equal to two and for lens spaces you also only have for ln plus 1n you have n copies of Q in dimension zero and n plus 1 copies of Q in every even dimension that's greater than or equal to two. Any more conversation requests? Any other questions? Is the right way to think about why you choose this formal variable that you're doing something analogous that you're a model for symplectic homology? Right so that's what you're doing but you've kind of been able to cheat a little bit so normally your differential would have like an infinite number of terms in it but because we could cheat and use a time independent almost complex structure and still be able to define the complex that's why we were able to drop the extra terms but that's pretty much exactly what's going on. Suppose you took two dynamically convex, just go back to the convex sum idea, you took two dynamically convex and then you did a convex sum but you did a really really really small convex sum where you had very precise control over this sort of, you had very precise control over this hyperbolic orbit in the finite energy place that would be sort of introduced, is there any chance that these ideas could extend in that case? I mean maybe but it's my understanding that even when you do a convex sum you always end up introducing a new... You do, but you know exactly where it is, you know pretty quickly you have pretty much generated longer ones. Yeah you could use action filtration arguments anyway. No because the, so my answer is no. So I mean it's conceivable that we could define some sort of linearized contact tomology if we, there's another type of building that we would have to consider and we'd have to be able to understand how to glue it or exclude curves limiting on it and you'd have to take some sort of like augmentation. So... I'm going to disagree with you on this one. We might be able to do it. Well that's, yeah I was saying you might be able to... Not linearized. Without, okay. I might not. Yeah. Anything is possible. Okay I think Homan's ready to clap for me and get coffee.