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      Questions tagged [sg.symplectic-geometry]

      Hamiltonian systems, symplectic flows, classical integrable systems

      9
      votes
      3answers
      278 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
      178 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
      297 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
      297 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 ...
      3
      votes
      0answers
      54 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\...
      4
      votes
      0answers
      68 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 ...
      1
      vote
      1answer
      161 views

      Floer equation and Cauchy Riemann equation

      Consider a symplectic manifold $(M,\omega)$ with the property that $\pi_2(M) = 0$. Given a time dependent hamiltonian $H_t$ on $M$, and a $\omega$-compatible almost complex structure J on M, we may ...
      2
      votes
      0answers
      87 views

      Limitations of the splitting construction and SFT

      I am trying to understand the so-called symplectic field theory (SFT) machinery used in symplectic topology. As I understand it, one of the applications of SFT (or rather, of the splitting ...
      2
      votes
      0answers
      62 views

      Transitivity of Diff on the space of embeddings of balls

      Given 2 symplectic embeddings $g_0$ and $g_1$ of a 4-ball of radius $r \leq 1$ into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from $\mathbb{R}^4$), does there exist ...
      3
      votes
      0answers
      75 views

      Large isometry groups of Kaehler manifolds

      Let $M$ be a closed simply-connected Kaehler manifold that is not isomorphic to a product of lower-dimensional Kaehler manifolds. Pick an orientation for $\mathbb{C}$; this endows $M$ with an ...
      4
      votes
      0answers
      67 views

      Pairs of J-holomorphic curves

      Let $(M, \omega)$ be a symplectic 4-manifold and let $A$ and $B$ be symplectic submanifolds on M such that $A \cap B = p \in M$.Under what conditions can I find a $\omega$-compatible almost complex ...
      3
      votes
      0answers
      108 views

      An inequality for symplectic manifolds

      Question: Is there a closed symplectic manifold satisfying the following? $$|Td(M)| > \sum_{i \in \mathbb{Z}} b_{i}(M) $$ here $Td(X)$ is the Todd genus of $M$ i.e. the integral of the Todd class ...
      3
      votes
      0answers
      103 views

      Symplectic Chern class of holomorphic symplectic manifold

      I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this ...
      4
      votes
      0answers
      106 views

      Diffeomorphism of $ \mathbb{C}P^2 \# ~\overline{\mathbb{C}P^2}$

      I am currently reading Dusa McDuff's paper "Blow ups and symplectic embedding in dimension 4" and had a few questions regarding the paper. In the paper McDuff uses the following notation. $X = \...
      5
      votes
      1answer
      181 views

      Multiple mirrors phenomenon from SYZ and HMS perspective

      There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...

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