13

I think a mainstream answer would be that symplectic geometry has two (seemingly opposiate, but actually related) aspects: rigidity and flexibility, it is the rigidity aspect that makes symplectic geometry a kind of geometry.
The study of the rigidity of symplectic manifolds dates back to Gromov's groundbreaking work on $J$-holomorphic curves, where $J$ is ...

9

This is not possible. There is something called the growth rate of symplectic cohomology which is subexponential for affine varieties and exponential for cotangent bundles of higher genus surfaces (amongst other things). This was proved by McLean:
https://arxiv.org/abs/1011.2542

8

I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.
1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in ...

5

As pointed out in the comments, the term geometry can have many meanings. Given a symplectic manifold $(M,\omega)$, I think almost (you never know!) all of us will agree that a Riemannian metric $g$ on $M$ would qualify as a geometric structure. While the symplectic form $\omega$ on its own does not provide a metric, the notion of an $\omega$-compatible or $...

4

The relative homology long exact sequence puts this group in between $H_2(\mathbb{CP}^n)=\mathbb{Z}$ and $\mathbb{Z}/2$. It maps surjectively to $\mathbb{Z}/2$. Let D be a generator for relative homology. Then $2D$ generates $H_2(\mathbb{CP}^n)$. The Maslov number of a relative class coming from $H_2(\mathbb{CP}^n)$ is twice the Chern class; for the positive ...

3

Let me try to prove an infinitesimal version where an area preserving diffeomorphism is replaced by a Hamiltonian vector field.
Let H be a function in the unit disc, constant on the boundary. Consider the Hamiltonian vector field $(H_y,-H_x)$ and the respective infinitesimal diffeomorphism
$(x,y) \mapsto (x+\epsilon H_y, y-\epsilon H_x).$
When is it ...

dg.differential-geometry at.algebraic-topology riemannian-geometry differential-topology sg.symplectic-geometry

2

This is an affirmative answer to $1$ (assuming by Hamiltonian you mean in the sense of real symplectic geometry).
Yes. Due to Sumihiro's theorem (+ standard facts about Hamiltonians). In fact the assumptions smooth quotient and symplectic are not necessary, just smooth and quasi-projective is enough. Throughtout when I mean symplectic I mean in the real ...

ag.algebraic-geometry dg.differential-geometry sg.symplectic-geometry poisson-geometry hamiltonian-mechanics

1

This bracket is sometimes called the Lagrange bracket in the literature (although that terminology is unfortunately not universal, and sometimes refers to something different). It can be characterised as the map $\lbrace\cdot,\cdot\rbrace:C^\infty(M)\times C^\infty(M)\to C^\infty(M)$ that satisfies $[X_F,X_G] = X_{\{F,G\}}$ (where $X_F$ is the vector field ...

1

I think that the best answer so far comes from this paper: https://arxiv.org/abs/1811.07830 , where it is proved that homotopy categories of dg-categories and various flavours of A-infinity categories are equivalent.

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