I like when you read through a chapter and think to yourself "huh this is pretty easy" and then you get stuck on the first exercise

@MikeMiller My understanding is that the answer is yes (even in the more general setting of complex deformation equivalence), has been understood for a while and is due to Kodaira. Note if b_2<8 then by Donaldson the form is automatically diagonizable.

@PVAL-inactive Right, I knew the latter, and $E_8$ is not allowed if $b_2 < 10$. All you could possibly have is $nH$ for $n < 5$.

@Mike Gompf-Mrowka cites Barth-Peters-Van de Ven as a place where Kodaira's classification is written out. I imagine the theorems are in that book.

Have you read that paper?

My advisor periodically writes theorems from complex surface theory on the board (without any proof ever) and almost always attributes Kodaira.

No.

I think my adviser probably knows this stuff without ever looking at any proof, so I wanted to find a source he probably got it from (as all the original Kodaira papers are infamously hard).

I probably want to read it at some point. But I'm happy to assume the relevant complex surface theory.

Apparently the paper was essentially done before they spoke to each other, with Gompf having already proven the topological things modulo the Seiberg-Witten theory and Mrowka having already computed the SW-theory stuff without any intended application.

or was that paper donaldson invariants. Looking at it now, I guess it was donaldson invariants

I think it's more Donaldson theory than SW. But most of the work seems very similar to what was published in Morgan Mrowka Ruberman (1994), so I imagine he had known it for a while.

Well Gompf-Mrowka was published in 1993 in the Annals (so probably submitted well before that).

Good point. Maybe he worked with $T^3$ and that gave the inspiration for MMR.

Also apparently written immediately following Mrowka's thesis if i recall the story Gompf told.

That's new to me.

When did SW-theory come along? Mid 90's

?

I guess 1994

94 I think

There wasn't really a satisfying version with boundary in any cases other than $S^3$ until the early 2000s

I proved that for any finite subgroup $G$ of $SO(2n) \cap Sp(2n,\Bbb R$ there is a symplectic structure on $\Bbb R^2n$ with symplectic mapping class group $G$. My advisor told me that maybe one day I'd be able to use that for something.

Witten's paper was in 94 according to the proof of the Thom conjecture.

Certainly the thom conjecture was using SW theory.

Yup.

@PVAL-inactive I find that really interesting. Have you written it down?

Also, is it known that you can realize any such group as a sympl mapping class group on a compact manifold?

@MikeMiller It's actually quite easy, but my advisers comment towards suggested to me that it probably wasn't that interesting. I'll probably tell other people about it, and if enough people tell me to write it down I'll write a note and post it. It's really just a sort of manipulation with the non-squeezing theorem.

The second question is probably known to be false for some subtle reason. I think constructions in this setting are probably much harder in the compact case.

Maybe you can tell me at some point. How badly exotic is the symplectic structure near infinity?

It's a pullback of a symplectic structure on some funnily embedded ball in R^4.

OK, so it can be weird at infinity. No hope of gluing these in as patches somewhere.

Basically choose a finite set in S^n-1 which G acts freely and transitively on and attach a ball along a thin tube along each ray emmenating from this finite set.

There's a natural action of G on this manifold by rotation, and this action descends to the symp. MCG. by gromov non-squeezing.

How do you prove that the MCG is at most G?

It should be sort of clever manipulation of the fact that the mcg of the standard ball is trivial.

OK.

I actually hadn't written up that direction, but I think it isn't too bad.

I'm a sucker for automorphism groups and mapping class groups in general.

The picture kind of looks the standard ball and and tube models for molecules.

I think I could see the picture.

why can $K_{3}$ not have a stable matching

i am so confused

Have you seen the proof that there is a compact hyperbolic 3-manifold with given finite isometry group? It's surgery trickery. I assume it wouldn't be very helpful to you, but there it is.

@MikeMiller This is kind of stemming from the Auckly-Kim-Melvin-Ruberman "Equivariant Corks" paper. Which has a similar construction in that world, and my advisor has some prelim. work where he translates those ideas into diffeotopy groups of exotic R^4's. So I was wondering if there was stuff using AKMR's kind of basic idea into symplectic or other worlds.

I saw Auckly talk about it, it was nice. I haven't read the paper yet though. A friend of mine posted a paper recently with a different proof that those actions give corks.

Tange?

Wong

It's a note.

I don't think I like how the AKMR paper is written, but its very short. I haven't actually read the sort of interesting part (the construction of a bunch of disjoint embeddings of a cork into some 4-manifold where the twists all give different 4-manifolds).

Though I imagine examples of that maybe were close to being known before that paper.

I didn't get a chance to look. It's not the sort of paper I get to spend much time on.

Actually I can probably write examples of that myself.

I had a new idea. I think it's probably workable. We'll see if my advisor shoots it down on Friday.

Can it prove something I'd care about?

Like for instance the ribbon-slice conjecture.

Probably not. The most applicable idea I ever had was completely unworkable.

Is it important in your field to actually have direct applications to topology and geometry? I see a lot of students publish things involving various Floer theories (this group intersects trivially with your advisers students probably) that don't seem to have any application to questions that don't explicitly have those Floer theories involved..

@PVAL-inactive It's important to me and to getting jobs. But sometimes you can write things with expected applications and not actual applications. Usually this takes the form of "Develop invariant and hope it defines new homology cobordism/concordance invariants".

Hello, can someone explaine me this

yes, can. will, who knows?

You start to get a sense for whether or not it's promising. I can look at Floer theory papers and have a decent guess about whether they will ever have interesting applications.

i don't understand "motivated by the functions u^p and v^p .." and he introduce the function $v=\Phi(u)$

@MikeMiller With Donaldson invariants you'd think you'd always get some geometric information with anything you do, as the invariants are somehow already seem to be defined with respect to interesting information about metrics.

I think the various ideas I've had are all promising for applications. I got into the field because I care about topology first. Gauge theory is just also interesting.

As an observer that is.

someone for help please ?

I guess I feel like questions which take place entirely in the symplectic category are interesting for their own sake, but I've definitely run into people who only care if they immediately imply something about 3 and 4 manifolds. It's very interesting to me when someone can prove something about 4-manifolds using contact and symplectic topology, but that certainly isn't the overwhelming reason I am interested in these things. I probably think Legendrian knots are just as interesting as knots.

@PVAL-inactive If someone can prove that a flavor of instanton homology is equal to a flavor of heegaard Floer I'm excited even though that will have very few "pure topology" consequences. If they can prove that I(M) = 0 => M = S^3 I'm interested. But on the other hand if I can define an invariant but I can't do anything with it, or even calculate it, who cares?

Whereas I know other people who are happy to have an invariant whether or not they can use it for anything.

So I think we have similar feelings, just replace symplectic stuff with gauge theory.

@MikeMiller Well it probably would show certain theorems based on Giroux would no longer need it. I am happy whenever that happens.

That's in response to your first statement.

I'm not sure off the top of my head what you're thinking of.

The relation ship between contact structures and Heegard floer is through Giroux. The relationship between contact structures and monopoles at least isn't.

Amusingly the relationship between Heegaard Floer and monopoles is also through Giroux, or at least I think Ko's proof uses open book decompositions.

Ko proved ECH=Heegard Floer right?

Is ECH= monopoles known?

hey

I guess the connection between ECH and Monopoles comes from studying the associated monopole to the symplectic form a la Taubes.

just a quick question if I have a group G or order pqr with p < q < r I would like to prove that it has a normal subgroup of either p,q, or r. I proved that it must have a normal subgroup if we assume it is not simple. Is there one direct way to prove this

without dividing into cases?

Both of those are known and both are independently due to Colin-Ghiggini-Honda and Kutluhan-Lee-Taubes.

The two proofs are pretty different. The theorem is just that they're isomorphic, natural isomorphism isn't known (so it should be true that it's open that the 4-manifold invariants agree, even though everyone believes it).

A proof that they're naturally isomorphic should go by creating some sort of cobordism maps on ECH inherent to contact/symplectic geometry and then extending one of the given proofs over these.

@PVAL-inactive Nevermind, you're right, ECH = SWFH was proved by taubes and cited away.

In probability, since expectation is linear, i.e. E[aX+bY]=aE[X]+bE[Y], is then also true that E[aX-1]=aE[X]-1 ?

Yes.

@MikeMiller Was that yes to me?

@MikeMiller I guess ECH is some homology theory derived from the SFT of the symplectization. I guess I am proving (have proved) things about the latter.

@litmus Yes.

Ok, thank you!

@PVAL-inactive Also yes. The generators are closed Reeb orbits and the differential counts holomorphic curves limiting to the Reeb orbits, counted in an appropriate way (analogous to the fancy counting you have to do for GW = SW).

The relationship of instantons to any of this is mysterious.

I wonder if the new Cioba-Wendl paper where they prove that if a contact manifold Y is a symplectic manifold has an exact symplectic cobordism to S^3, then it has a contractible Reeb orbit says in terms of the Heegard floer homology of Y.

I'm not sure if contractible orbits behave in a special way inside ECH.

My roommate would probably know.

@MikeMiller Hello

Would like to ask you about your comment in one of my questions(deleted)?

Howdy

Ping me whenever you are available

All right, I'm gonna ask a bit of a philosophical question. Is math discovered or invented?