# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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### (symplectic) $h$-cobordism from $S^1\times S^2$ to itself

I ran into an oriented smooth $h$-cobordism from $S^1\times S^2$ to itself in my project. I wish to argue that it is diffeomorphic/homeomorphic to the product. From this question 4-dimensional h-...
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### Existence of Liouville vector fields on symplectic manifolds

Let $(M, \omega)$ be a symplectic manifold. A vector field $V: M \to TM$ is Liouville if $L_{X} \omega=\omega$. The existence of a Liouville vector field implies that $(M, \omega)$ is exact: the one-...
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### Translation of Marsden-Weinstein-Meyer into classical mechanics language

The Marsden-Weinstein-Meyer theorem is expressed in a too general way to be understood by a mean square physicist, as me. However, if we limit the scope to a Hamiltonian mechanics, it should be ...
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### Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the ...
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### An extension of symplectomorphism group

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$. We consider the ...
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### Conley Zehnder index for Floer homology of a symplectomorphism

I'm trying to get some intuition for the Conley-Zehnder index in the setting of Floer homology of a symplectomorphism $\phi : (M,\omega) \to (M,\omega)$. Let's assume that $\phi$ only has non-...
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### How are Poisson brackets and the variational principle related?

In the lecture Space and spaces, Segal argues that the origin of non-commutativity in classical mechanics “which is encoded in the Poisson Bracket” is the fact that the evolution of classical states ...
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### How not to use J-holomorphic curves [closed]

The field of symplectic topology is filled with subtle traps for the unwary, particularly when it comes to the analysis of $J$-holomorphic curves. So that the next generation of symplectic topologists ...
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### Differential geometry of Donaldson-Thomas invariants

The Donaldson-Thomas invariants are defined by Thomas in the paper A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, following the proposal in Gauge theory in higher ...
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### Mirzakhani's hyperbolic method generalized to moduli space of stable maps

I've been learning about Mirzakhani's use of hyperbolic geometry to compute Weil-Petersson volumes of moduli space of curves, and the application to proving Virasoro constraints for a point. Why have ...
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### Proove the equivalence of two Lagrangians

Consider Lagrangian $$L_1= q\dot{\alpha}+\alpha^2$$ The Euler-Lagrange equations for $q$ and $\alpha$ read $$\dot{q}=2\alpha$$ $$\dot{\alpha}=0$$ These two equations can be combined, ...
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### Two questions on history of symplectic geometry in the 80's

I have a question about the history of two important results from the eighties in symplectic geometry. In both cases it seems that important results were developed (almost) simultaneously by ...
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### Isomorphism of certain irreducible representations over finite fields

We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V)$ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
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### Organizing mirror pairs

At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...
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### Infinity local systems

I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems". From what I've been told, given a good cover $\{U_i\}$ of $X$, ...
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### Derivation of an uncertainty principle from the symplectic non-squeezing theorem

Is there a derivation of an uncertainty principle or uncertainty-type principle from the symplectic non-squeezing theorem?
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### What is symplectic rigidity?

What is an explanation for what the theory of symplectic rigidity is and what kind of questions it can answer? I was led to this after reading about the symplectic non-squeezing theorem of Gromov.
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### 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 ...
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### Representable functors and symplectic co-tangent bundles

I've been banging my head against something that I feel should follow from abstract non-sense, and I hope someone here can set me straight. Let $\mathcal{M}$ be the category of smooth manifolds, with ...
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