# Questions tagged [symplectic-geometry]

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

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### Why should we introduce symplectic width？

I'm a beginner in Symplectic topology and I'm learning the book Introduction to Symplectic Topology. In the second chapter of this book, they introduce the notation symplectic width $w_L(A)$ for ...
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### Conditions on space of smooth almost complex structures so that it's a banach manifold

In the following paper by A.Abbondandolo and M. Schwarz https://arxiv.org/pdf/math/0408280.pdf in section $1.6$ we consider the set of smooth almost complex structures $\mathcal{J}$ on $T^*M$ ...
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### What does it mean by "A continuous family of symplectic forms"

I have difficulty in understanding the meaning of "A continuous family of symplectic forms". I have seen this in many papers on symplectic geometry. Does it mean, we have a one parameter ...
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### A variety is the moduli space of structure sheaves of points

In the last paragraph of the first page of this paper, it is mentioned that an $n$-dimensional Calabi-Yau manifold $X$ is the moduli space of structure sheaves of its points and I am not really sure ...
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### $\omega$ is a symplectic form then $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ is injective for all $q\leq n-k$ and surjective for all $q\geq n-k$

Let $(M^{2n},\omega)$ be a symplectic manifold of dimension $2n$. Let $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ be the map given by $L^k_{\omega}(\alpha)=\alpha\wedge\omega^k$. Then is it true ...
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### Explicit evaluation of symplectic form on cotangent bundle

Let $M$ be a manifold and denote by $T^*M$ its cotangent bundle. Let $(x,U)$ be a coordinate chart so that $x: U\to \mathbb{R}^{n}$. Let $p\in U$ and $v\in T^*_pM$, then we can write $v = v_i dx^i$ ...
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### Condition for a Lagrangian submanifold to be totally geodesic

Consider $(M,\omega)$ a symplectic manifold and $L\subset M$ a Lagrangian submanifold. Let $J$ be an almost complex structure and $g:=\omega(.,J.)$ a compatible riemannian metric . Are there ...
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### How to learn math after your PhD is finished [closed]

Question: How does someone go about learning advanced topics in Math after they're done with their PhD? Specific example: You've done your undergrad and masters degrees in math and learned from ...
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Let $(M,\omega)$ be a (symplectic) manifold. I want to compute the Maslov index of a loop $\gamma:\mathbb{R}\to M$ directly. In order to do that I have to find a (symplectic) trivialization of $\gamma^... • 121 5 votes 1 answer 87 views ### volume preserving version of Moser's theorem There exists an well-known theorem of Moser : Thereom(Moser) Let$M$be a compact oriented smooth manifold and$\alpha,\beta$be volume forms whose total volumes are the same. Then, there exists a ... • 337 1 vote 0 answers 52 views ### "Appropriate" Hamiltonian function of simple pendulum Consider a simple pendulum of length$\ell$and mass$m$, where the only force is gravity. If$\theta$is the angle between the rod and the vertical direction, and$\xi$is the coordinate along fibers ... • 1,232 3 votes 1 answer 39 views ### Bundle isomorphisms for$J$-holomorphic tangent bundle In Chapter 14 of Lectures on Symplectic Geometry by da Silva, she claims that, if$(M,J)an almost complex manifold, then there are real bundle isomorphisms \begin{align*}\pi_{1,0}:TM\otimes\mathbb C&... • 1,232 4 votes 0 answers 84 views ### Proving that implicit midpoint method for Hamiltonian systems is symplectic using a criterion The implicit midpoint rule is defined as $$y_{n+1}=y_n+hJ^{-1}\nabla H\left(\frac{y_{n+1}+y_n}{2}\right).$$ wherey=(p,q)$. I know how to prove that this method is symplectic by hand, using the ... • 1,377 1 vote 1 answer 35 views ### Clarification on the definition of a symplectic integrator According to the notes that I am reading, a numerical one-stop method$y_{n+1}=\Phi_h(y_n)$is said to be symplectic if, when applied to a Hamiltonian system, the discrete flow$y\mapsto \Phi_h(y)$is ... • 1,377 0 votes 0 answers 15 views ### Understanding a condition for a transformation to be symplectic I am reading these notes to learn about symplectic mapping, and there is something that I don't understand. Theorem 5 on page 11 says: Let$(p,q)\rightarrow (P,Q)$be a smooth mapping, close to the ... • 1,377 0 votes 0 answers 47 views ### The Fubini-Study form on the complex projective space can be obtained from the Symplectic reduction For a complex complex plane$\mathbb{CP}^{n}$, there is a natural Kähler form $$\omega=\partial\overline{\partial}\log\sum_{i=0}^{n}z_i\overline{z_i}$$ Which is so called the Fubini-Study form, where$...
I'm interested in methods for numerically integrating Hamiltonian systems \begin{align} \dot q & = +\frac{\partial H}{\partial p} \\ \dot p & = -\frac{\partial H}{\partial q} \end{align} ...
I am trying to find a canonical transformation for this Hamiltonian: $$H(q,p) = p + e^p\cos(q)$$ and I'm trying to find a conjugate momentum $P$ to the variable $Q$ defined ...