# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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### 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**

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### 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**

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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 ...

**5**

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**1**answer

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 ...

**9**

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**2**answers

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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### 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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### 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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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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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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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 ...

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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 [...

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**2**answers

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**

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**1**answer

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 ...

**3**

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**2**answers

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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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 ...