Hamiltonian systems, symplectic flows, classical integrable systems

**14**

votes

**1**answer

344 views

### Are symplectic methods used in (classical) Economics?

The tl;dr question is this: are economists using coordinate-free formulations in studying theory?
Borrowing from classical mechanics, the framework I have in mind for classical economics--involving ...

**0**

votes

**1**answer

82 views

### Moment maps and flat degenerations of toric varieties

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...

**4**

votes

**1**answer

155 views

### Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to
$8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**3**

votes

**1**answer

188 views

### What are the indecomposable classes on a del-Pezzo surface?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**4**

votes

**2**answers

157 views

### actual dimension of concrete moduli space of holomorphic curves vs its virtual dimension

I am looking at exercise 6.3.3 in Mcduff's and Salamon's book J-holomorphic curves and Symplectic topology, which basically gives an example of a moduli space whose actually dimension is greater than ...

**3**

votes

**1**answer

93 views

### neighborhood of symplectic surfaces

I want to know if there is a uniqueness (in any sense) theorem for the symplectic structure in a neighborhood of a symplectic surface in a four dimensional symplectic manifold. Or more generally for ...

**5**

votes

**1**answer

180 views

### Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold

Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing ...

**4**

votes

**1**answer

165 views

### Learning Quantum (Co)Homology and Landau Ginzburg Superpotential

I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer ...

**1**

vote

**1**answer

115 views

### Contact structures on circle cross plane

Can anyone provide an explicit contactomorphism between the following two contact structures on the circle cross the plane?
1) The standard contact structure on threespace, but with the line that ...

**2**

votes

**2**answers

122 views

### Hamiltonian group actions in the context of holomorphic line bundles

When studying Hamiltonian group actions, a very nice set up might be to take the following:
Let $M$ be a compact Kähler manifold with (integral) Kähler form $\omega$, endowed with a Hamiltonian ...

**1**

vote

**1**answer

127 views

### Show that the symplectic action 1-form on loop space is closed

I am struggling to see how the symplectic action 1-form $\alpha_H$ on the loop space of a symplectic manifold $(M,\omega)$ is closed. It is defined by
$\alpha_H(Y) = \int_0^1 ...

**0**

votes

**0**answers

91 views

### Floer homology for manifolds with contact boundary

I am reading the paper " A survey of floer homology for manifolds with contact boundary" by A. Oancea. In theorem 2.1, he discussed the invariance of
Viterbo's theory of symplectic homology with ...

**8**

votes

**0**answers

76 views

### Kernel of “Hat to Plus” in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$
induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes.
Given a ...

**12**

votes

**2**answers

257 views

### Is there a relationship between Fourier transformations and cotangent spaces?

Maybe a trivial question but I can't seem to find it treated anywhere.
Consider a smooth manifold $Q$ (configuration space) and its cotangent bundle $T^*Q$ (phase space). Any function ...

**1**

vote

**0**answers

68 views

### AKSZ sigma models for higher spin

The AKSZ framework constructs 2D sigma models in the BV formalism. Is there a generalization of the AKSZ approach to higher spin?

**0**

votes

**1**answer

84 views

### Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...

**1**

vote

**1**answer

207 views

### symplectic reduction for pair $(M,D)$

Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?

**0**

votes

**0**answers

60 views

### Convenient Basis Presentation of Lefschetz Decomposition

Let $V$ be an almost-complex vector space, equipped with a symplectic element $\omega \in V^{(1,1)}$. In terms of a basis $b^+_i \in V^{(1,0}$, $b^-_i \in V^{(0,1}$, does there exist a "simple" ...

**0**

votes

**0**answers

65 views

### Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?

Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...

**4**

votes

**1**answer

101 views

### Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question.
Is ...

**3**

votes

**0**answers

171 views

### Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, ...

**2**

votes

**0**answers

80 views

### Generation of compact Lagrangians over fields with characteristic 2

Let $\pi:X\rightarrow\mathbb{C}$ be a Lefschetz fibration, and assume that the fibers are exact or monotone. A classical result of Seidel says that all the closed weakly unobstructed Lagrangian ...

**2**

votes

**0**answers

68 views

### Is there an action functional for the s-dependent Floer equation?

The usual Floer equation (in local coordinates)
\begin{equation*}
\partial_su+J(t,u)(\partial_tu-X_{H_t}(u))=0
\end{equation*}
is derived as the gradient flow of the symplectic action functional ...

**3**

votes

**2**answers

162 views

### Momentum a cotangent vector

Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.
Furthermore, many sources ...

**4**

votes

**2**answers

929 views

### A question about Marsden-Weinstein reduction theory

Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J \colon M\to \frak g^*$ be its moment map then the ...

**3**

votes

**1**answer

104 views

### Reference of $\hbar$-differential operator from symplectic geometry perspective

I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...

**2**

votes

**0**answers

119 views

### Kahler identities on almost Kahler manifolds

Suppose that $A$ is a unitary connection on a Hermittian differentiable vector bundle $E$ over a Kahler manifold $X$, then we have operators $$\bar{\partial}_A: \Omega_{X}^{p,q}(E)\to ...

**4**

votes

**2**answers

386 views

### Symplectic boundary

Is there any work on manifolds (perhaps contact) with symplectic boundary (not asking about the boundary of a symplectic manifold)?

**2**

votes

**0**answers

89 views

### Symplectic form/Kahler metric on a toric manifold

I have a standard question about symplectic forms on toric manifolds:
Let $P$ be an $n$-dimensional Delzant polytope and let $X_P$ be the corresponding symplectic toric manifold with symplectic form ...

**2**

votes

**0**answers

20 views

### When does a symmetric Poisson manifold decompose into homogeneous pieces?

When people study representations, they pay a lot of attention to the irreducible ones. This is justified by the fact that, roughly speaking, every unitary representation decomposes into a direct ...

**6**

votes

**0**answers

190 views

### Schoenflies and symplectic topology

The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the ...

**25**

votes

**0**answers

788 views

### Why are there so few quaternionic representations of simple groups?

Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...

**0**

votes

**1**answer

55 views

### Principal bundle associated to a Courant algebroid

A Courant algebroid is defined as a real vector bundle equipped with a product and a symmetric bilinear on its space of sections satisfying a particular set of conditions. What would be the definition ...

**6**

votes

**5**answers

391 views

### In a contact manifold, is every tranverse 1-foliation given by some Reeb vector field?

Let $M$ be a contact manifold, and let $F$ be an oriented 1-dimensional foliation that is transverse to the contact structure.
Is there a contact form $\alpha$ whose associated Reeb vector field ...

**1**

vote

**1**answer

139 views

### Symplectic reduction: from indefinite signature to Riemannian signature

Let $(M,g,\omega)$ be a $d$-dimensional manifold equipped with a metric $g$ of signature $(t,s)$, $d = t+s$, and a symplectic form $\omega$. Let us assume that a Lie group $G\subset Isometries(M,g)$ ...

**8**

votes

**1**answer

268 views

### Can a symplectic manifold be recovered from its Lagrangians?

Something I have wondered idly about from time to time is:
If $(M,\omega), (M',\omega')$ are symplectic manifolds, and you "know what the Lagrangians $L \subset M$ resp. $L' \subset M'$ are," can ...

**3**

votes

**0**answers

194 views

### Complex symplectic reduction

Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...

**3**

votes

**2**answers

264 views

### Boundary geometry of a contact manifold

Let $(M, \xi = \text{ker}\,\alpha)$ be a compact contact manifold with non-empty boundary. Vaguely asked, is there any natural geometric structure on the boundary $\partial M$ induced from the contact ...

**9**

votes

**3**answers

1k views

### Reasons for the Arnold conjecture

I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, ...

**1**

vote

**0**answers

206 views

### Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$
I am looking for a proof for ...

**2**

votes

**0**answers

222 views

### Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...

**1**

vote

**2**answers

351 views

### Semistability in GIT

If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose ...

**0**

votes

**1**answer

92 views

### Is the group $Ham(M,\omega)\cap Iso_{0}(M,g)$ compact?

let $(M,J,g,\omega)$ be a compact K\"ahler manifold of complex dimension at least $2$. As usual $J$ is the complex structure, $\omega$ is the symplectic form, $g$ is the Riemannian metric and
...

**13**

votes

**2**answers

1k views

### Symplectic formulation of statistical physics

Does there exists a symplectic formulation of statistical physics?
I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated ...

**0**

votes

**1**answer

222 views

### On Gromov's Theorem on Symplectic Homotopy

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...

**4**

votes

**3**answers

691 views

### Conformal-symplectic geometry ?

I think in priciple it's possible to consider a theory of "conformal-symplectic manifolds", in an analogous fashion as the usual conformal geometry.
To spell out the spontaneous definitions: say ...

**0**

votes

**1**answer

185 views

### Some quantities which definitions are (somehow) similar to the classical Divergence

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...

**2**

votes

**1**answer

160 views

### Symplectic (contact) structure on $M_{n}(\mathbb{R})$

Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under ...

**1**

vote

**1**answer

119 views

### Reeb orbit and open books

Weinstein conjecture is about existence of a closed orbit of the Reeb vector field on every contact manifold. On the other hand, we know every contact 3-manifold admits a compatible open book, which ...

**2**

votes

**1**answer

232 views

### An identity for Futaki-Donaldson invariant

Let $(X,L)$ be a polarized projective variety
Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :
a scheme $\mathfrak X$ with a $\mathbb C^*$-action
a ...