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        Questions tagged [sg.symplectic-geometry]

        Hamiltonian systems, symplectic flows, classical integrable systems

        2
        votes
        0answers
        75 views

        Mixed characteristic in symplectic geometry

        Are there any mixed-characteristic phenomena in symplectic geometry/mirror symmetry? There are papers on symplectic geometry by Abouzaid (inspired by Kontsevich--Soibelman, I believe) in which there ...
        4
        votes
        0answers
        66 views

        How is the instanton Floer homology of Seifert fibrations related to that of a trivial fibration

        My question centers around the relationship of the Chern-Simons theories of a Seifert fibration and the trivial product space $\Sigma_g \times S^1$, and its implication for instanton Floer homology. ...
        4
        votes
        0answers
        54 views

        Contact 3-manifolds with hyperkahler Stein fillings?

        Is there any classification result on (homeomorphism type) of contact 3-manifolds $\Sigma$ that have Stein filling $W$ that is 1. Hyperkahler (s.t. Stein structure is the Kahler part of it) 2. not ...
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        votes
        1answer
        197 views

        Relationship between Gromov-Witten and Taubes' Gromov invariant

        Fix a compact, symplectic four-manifold ($X$, $\omega$). Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; \mathbb{Z})$ defined by weighted counts of pseudoholomorphic ...
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        votes
        2answers
        238 views

        Symplectic equivalent of commuting matrices

        It is well known what happens if two real symmetric matrices commute, i.e. if we have two matrices $A$ and $B$ such that $A=A^T$, $B=B^T$ and $AB=BA$. The answer is given in terms of diagonalization: ...
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        votes
        0answers
        110 views

        Mapping classes as Lefschetz fibrations over surfaces with positive genus

        Let $\Sigma_{g,r}$ be the surface of genus $g$ and $r$ boundary components. It is known that, from a positive factorization of a mapping class $\phi$ in the mapping class group $MCG(\Sigma_{g,r}, \...
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        vote
        0answers
        94 views

        Alternative Lie bracket on $\chi^{\infty}(M)$ where $M$ is a Riemannian manifold with a symplectic structure

        Let $(M,\omega, g)$ be a Riemannian symplectic manifold. The Poisson bracket of two functions $f,g$ is denoted by $\{f,g\}$. The Hamiltonian vector field and gradient vector field ...
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        votes
        0answers
        81 views

        Maslov index of pair of paths in $\mathcal{L}(2n)$ and its relation with the Maslov index of a loop in $\mathcal{L}(2n)$

        I'm reading [RS] and I was wondering what kind of connection there is between the Maslov index for a pair of paths $\lambda_0,\lambda_1 \colon [a,b] \to \mathcal{L}(2n)$ as defined in [RS] and the ...
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        votes
        0answers
        116 views

        Introduction to the Adler-van Moerbeke theory

        Is there a good introduction to the Adler-van Moerbeke theory of solving completely integrable systems by linearizing the flow on the Jacobian of an algebraic curve, for someone with a background in ...
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        votes
        3answers
        331 views

        Why are Lagrangian subspaces in a symplectic vector space interesting?

        A subspace in a symplectic vector space could be one of two extremes: either symplectic (meaning the form is nondegenerate there) or Lagrangian. Or it could be something between the two, meaning a ...
        6
        votes
        1answer
        192 views

        Strip breaking phenomenon in the Gromov compactification of Moduli space of Pseudoholomorphic curves

        As the title suggests, I want to understand the strip breaking phenomenon that happens when I Gromov-compactify the moduli space of pseudoholomorphic curves from the holomorphic strip $\Bbb R \times [...
        7
        votes
        2answers
        310 views

        Volume of manifolds embedded in $\mathbb{R}^n$

        Let $N$ be a closed, connected, oriented hypersurface of $\mathbb{R}^n$. Such a manifold inherits a volume form from the usual volume from on $\mathbb{R}^n$ and has an associated volume given by ...
        8
        votes
        1answer
        306 views

        Constants of motion for Droop equation

        There is an important ODE system in biochemistry, Droop's equations: $$s'=1-s-\frac{sx}{a_1+s}$$ $$x'=a_2\big(1-\frac{1}{q}\big)x-x$$ $$q'=\frac{a_3s}{a_1+s}-a_2(q-1)$$ Relatively easy one finds a ...
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        votes
        2answers
        149 views

        Symplectic vector fields everywhere transverse to a co-dimension one hypersurface

        Usually when speaking about vector fields transverse to a hypersurface in a symplectic manifold, we talk about Liouville vector fields, i.e. vector fields $X$ with the property that $\mathcal{L}_X\...
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        votes
        0answers
        70 views

        Lagrangian subgroup of a nonabelian Lie group

        My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory. See a previous post for other background ...

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