MathOverflow Stack Exchange
MathOverflow Stack Exchange

Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

1430 questions
42
votes
1 answer

Is a symplectic camel actually prohibited from passing through the eye of a needle?

Gromov's symplectic nonsqueezing theorem asserts that in the symplectic space ${\bf R}^{2n}$ with canonical coordinates $p_1,\dots,p_n,q_1,\dots,q_n$, and two radii $0 < r < R$, it is not possible to symplectomorphically map the ball $B(0,R)$ into…
Terry Tao
  • 108,865
  • 31
  • 432
  • 517
24
votes
4 answers

Reasons for the Arnold conjecture

I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, then we can consider the Hamiltonian equation of…
nicolas
  • 683
18
votes
7 answers

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 $M$. I believe that a Lagrangian fibration of $M$ is a collection of…
Theo Johnson-Freyd
  • 52,873
16
votes
2 answers

Is every symplectic manifold a Hamiltonian reduction of a cotangent bundle?

Today I heard the claim that in practice, all symplectic manifolds that people care about arise as the Hamiltonian reduction of a cotangent bundle $T^{\ast}(M)$ under the action of a Lie group $G$ ($M$ and $G$ may both be infinite-dimensional in…
Qiaochu Yuan
  • 114,941
14
votes
1 answer

Why believe Kontsevich cosheaf conjecture?

Kontsevich cosheaf conjecture roughly states that wrapped Fukaya category can be recovered from local information on the Lagrangian skeleton. What are some reasons why would one believe it? I believe it looks most reasonable in the microlocal…
rori
  • 149
14
votes
0 answers

To what extent does Floer cohomology detect Hamiltonian non-displaceability of immersed curves?

Floer cohomology for immersed Lagrangians is introduced by Akahi, Manabu; Joyce, Dominic, Immersed Lagrangian Floer theory, J. Differ. Geom. 86, No. 3, 381-500 (2010) and its one-dimension version (in the absence of "teardrops") is developed in…
Chris Woodward
  • 1,618
  • 11
  • 18
12
votes
2 answers

Special Hamiltonian diffeomorphisms

Is there any obstruction that prevents a Hamiltonian diffeomorphism on some symplectic manifold to be realized as the time-one map of the Hamiltonian flow of an autonomous Hamiltonian? In the same spirit, is there any obstruction that prevents a…
Marco Mazzucchelli
  • 523
11
votes
1 answer

Can a symplectic manifold be recovered from its Lagrangians?

Something I have wondered idly about from time to time is: If $(M,\omega), (M',\omega')$ are symplectic manifolds, and you "know what the Lagrangians $L \subset M$ resp. $L' \subset M'$ are," can you determine whether $M$ and $M'$ are…
Nathaniel Bottman
  • 1,569
11
votes
3 answers

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 to at least associate a fundamental class to the moduli space in…
Orbicular
  • 2,875
11
votes
3 answers

what prevents a manifold to be symplectic?

Are there any obstructions known which prevent an even dimensional orientable manifold from being symplectic? I am a novice in this area so I unfortunately I cannot make the question more precise. What I have in mind is raw and is as follows:from…
nikita
  • 1,335
  • 9
  • 18
8
votes
0 answers

Examples of symplectic manifolds whose Betti numbers are not non-decreasing

I am looking for examples of closed symplectic manifolds $(M,\omega)$ whose Betti numbers do not satisfy a non-decreasing property. Meaning, it fails to satisfy $b_k(M) \leq b_{k+2}(M)$ for some $k < n=\frac{1}{2}\dim M$. (Edit: I've been told in…
inkievoyd
  • 508
8
votes
2 answers

How can I tell whether a Poisson structure is symplectic "algebraically"?

My primary motivation for asking this question comes from the discussion taking place in the comments to What is a symplectic form intuitively?. Let $M$ be a smooth finite-dimensional manifold, and $A = \cal C^\infty(M)$ its algebra of smooth…
Theo Johnson-Freyd
  • 52,873
7
votes
3 answers

Symplectic structures on $M \times S^{2n}$

For $n > 1$, $2n$-dimensional sphere $S^{2n}$ does not admit symplectic structures. Then how about the product with a manifold? Are there any results about the symplectic structures on $M \times S^{2n}$?
Hwang
  • 1,388
7
votes
3 answers

look into Delzant Polytope

A Delzant polytope in R^n by definition is a simple, rational, and smooth convex polytope in R^n (Ana Cannas da Silva's book for notions.) Do you guys have any insight of the definition, for example, anything we can say about the shape? They satisfy…
Wayne
  • 377
7
votes
6 answers

Why are the following varieties symplectomorphic?

I saw a statement somewhere that for the Hirzebruch surfaces $F_n:=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}\oplus\mathcal{O}(n))$, $F_n$ and $F_m$ are symplectormorphic when $m$ and $n$ have the same parity. My question is: Why is this true? I can see…
Yuhao Huang
  • 4,982
1
2 3 4 5 6