Tagged Questions

7
votes
5answers
209 views

Does the Baker-Campbell-Hausdorff formula hold for vector fields on a (compact) manifold?

Consider a compact manifold M. For a vector field X on M, let $\phi_X$ denote the diffeomorphism of M given by the time 1 flow of X. If X and Y are two vector fields, is $\phi_X \ …
9
votes
3answers
211 views

To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?

This is probably a very elementary question in symplectic geometry, a subject I've picked up by osmosis rather than ever really learning. Suppose I have a symplectic manifold $M$. …
5
votes
4answers
295 views

How to see the Phase Space of a Physical System as the Cotangent Bundle

Two things today motivated this question. First, the professor said that in a lecture Thurston mentioned Any manifold can be seen as the configuration space of some physical s …
6
votes
2answers
160 views

Is the ‘massive’ Calogero-Moser system still integrable?

Background The (rational) Calogero-Moser system is the dynamical system which describes the evolution of $n$ particles on the line $\mathbb{C}$ which repel each other with force p …
4
votes
5answers
283 views

Understanding moment maps and lie brackets

I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is G with lie algebra g, acting on the symplectic manifold (M,w) by symplectomorphisms). I'm hav …
5
votes
3answers
209 views

Do there exist small neighborhoods in a classical mechanical system without pairs of focal points?

The question I will ask makes sense in much more generality, but I will leave the translation to the experts, since I'm only looking for a special case (and it would not surprise m …
6
votes
3answers
241 views

cotangent bundle symplectic reduction and fibre bundles

Suppose a compact Lie group $G$ acts on a manifold $M$ with only one orbit type $G/H$ ($H$ denotes the stabiliser group). Then the manifold $M$ becomes a fibre bundle over the quot …
1
vote
1answer
74 views

Flux homomorphism for manifolds with boundary

Hi all, I am wondering whether someone has considered the definition of the flux homomorphism for manifolds with boundary. More specifically, I am looking at the annulus and I wa …
11
votes
2answers
289 views

Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$. I foun …
2
votes
1answer
109 views

Is a Poisson Group a group object in the category of Poisson Manifolds?

I realized that I am very confused about a certain sign in the definition of a Poisson group. I will give some definitions, and then point out my confusion. Definitions Group ob …
2
votes
1answer
134 views

Different definitions of Novikov ring?

Following, e.g., Wikipedia's definitions, the (small) quantum cohomology ring of $X$ is defined over a "Novikov ring" consisting of formal power series of the form $$ \sum_{\beta \ …
2
votes
1answer
120 views

Homotopy classes of complex bundle maps and isotropic immersions into contact manifolds

This is a follow-up question to my previous one where I was trying to understand the classes of Legendrian immersions of circles into contact manifolds. I'm interested in classify …
5
votes
2answers
147 views

Kuranishi structures vs polyfolds

Moduli spaces of pseudoholomorphic curves do not carry the structure of a (compact) differentiable manifold in general (due to transversality issues). Nevertheless one would like t …
3
votes
2answers
226 views

Are Fukaya categories Calabi-Yau categories?

Let X be a compact symplectic manifold. There is an idea, I think probably originally due to Kontsevich, that we should be able to get Gromov-Witten invariants of X out of the Fuka …
6
votes
4answers
320 views

Has anything precise been written about the Fukaya category and Lagrangian skeletons?

At some point in this past year, some Fukaya people I know got very excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons." As I understand it, a …

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