Hamiltonian systems, symplectic flows, classical integrable systems

**20**

votes

**0**answers

570 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 ...

**13**

votes

**0**answers

302 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...

**11**

votes

**0**answers

356 views

### Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...

**11**

votes

**0**answers

618 views

### Compact Symplectic Fano (strongly monotone) manfiolds

What are known examples of compact symplectic Fano manifolds, apart from those that come from algebraic geometry?
We define symplectic Fano manifold as a symplectic manifold $(M,w)$, such that
...

**10**

votes

**0**answers

328 views

### flexibility of almost contact ``Reeb'' vector fields

New version of the question:
Given an odd dimensional manifold $V$, an almost contact structure is a pair of $(\alpha, \omega)$, where $\alpha$ is a non-vanishing 1-form and $\omega$ is a 2-form ...

**9**

votes

**0**answers

334 views

### Wrapped Fukaya categories of Stein manifolds

By the work of Abouzaid, we know that the wrapped Fukaya category of $T^\ast Q$ with $Q$ a closed smooth manifold is generated by a cotangent fiber. Basically, this is an application of Abouzaid's ...

**9**

votes

**0**answers

372 views

### Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...

**9**

votes

**0**answers

333 views

### State of research in moduli space of flat connections

I am a recent PhD student trying to settle into a research topic. Even though I have a current project I am working on, I am not particularly enjoying it and would like to switch. Before braving the ...

**9**

votes

**0**answers

445 views

### The open problem of nth quantization

In trying to explain a quote by E. Nelson, "First quantization is a mystery, but second quantization is a functor!" Baez points out what follows (full text available in this week find; I'm also ...

**9**

votes

**0**answers

634 views

### Kontsevich's lecture at Rutgers (1996)

Reading some papers about (homological) mirror symmetry I have found the reference to the unpublished Kontsevich's lecture at Rutgers University (Nov 11, 1996). I would like to know if someone has ...

**9**

votes

**0**answers

1k views

### Is Witten's new method of quantization useful for geometric complexity theory?

The Kempf-Ness theorem (see e.g. arXiv:0912.1132) - that the algebraic quotient of geometric invariant theory is also a symplectic quotient - suggests (to me) that certain physical constructions used ...

**8**

votes

**0**answers

227 views

### Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if
$$ du \circ j = J \circ ...

**8**

votes

**0**answers

469 views

### Pseudocycle definition of open Gromov-Witten invariants

I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as ...

**8**

votes

**0**answers

435 views

### SFT gluing on chain level in Floer homology?

I came across a new type of gluing in the FOOO paper http://arxiv.org/abs/1002.1660 (maybe not so new to experts). The situation is as follows: one considers a relative (to lagrangian) class and ...

**7**

votes

**0**answers

201 views

### From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine.
Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...

**7**

votes

**0**answers

376 views

### Degenerate moduli spaces in Floer homology

Let $(W,\omega)$ be a closed symplectially aspherical symplectic
manifold, and fix a Hamiltonian $H\in C^{\infty}(W\times S^{1};\mathbb{R})$
and a compatible almost complex structure $J$ on $W$. Given ...

**6**

votes

**0**answers

192 views

### Question on Ionel and Parker's paper: Relative Gromov Witten Invariants

In defining Gromov-Witten invariants using symplectic geometry, most of the trouble is to achieve transversality for moduli spaces of pseudo-holomorphic curves which are multiple covers of simple ...

**6**

votes

**0**answers

185 views

### Different complexifications of a real analytic Riemannian manifold

Hi,
I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well knwon fact that in a neighbourhood $U$ of the ...

**6**

votes

**0**answers

463 views

### Are conical symplectic resolutions Mori dream spaces?

This is one of these questions where it's tempting to just leave it at the title, but let me try to define the objects in question.
A conical symplectic resolution is a projective resolution of ...

**6**

votes

**0**answers

397 views

### How to prove that a certain action is hamiltonian?

I posted this question here on math.stackexchange.com. I have not had an answer, and I thought it could be more appropriate here.
(Please, If you judge this my opinion is wrong, then I will delete ...

**6**

votes

**0**answers

375 views

### Weakest condition for an integrable, almost-symplectic manifold?

I was recently speaking with someone who works in Computational Chemistry and they mentioned that in a lot of the computational simulations they do, they have systems that are not symplectic but still ...

**5**

votes

**0**answers

130 views

### Unobstructed Lagrangian tori

Let $X$ be a compact symplectic manifold, $U$ a Darboux chart, and $L$ a standard Lagrangian torus in $U$. Is $H^1(L)$ weakly unobstructed in the sense of Fukaya-Oh-Ohta-Ono (that is, does $b \in ...

**5**

votes

**0**answers

272 views

### Laplacians associated to symplectic cohomologies

I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...

**5**

votes

**0**answers

260 views

### Sophus Lie on the symplectic foliation theorem

Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e.
$$\mathcal ...

**5**

votes

**0**answers

383 views

### Jones Polynomial and Quantum Field Theory

I am trying Witten's paper but unable to re-produce the computations presented in the paper.
I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...

**5**

votes

**0**answers

258 views

### Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...

**5**

votes

**0**answers

456 views

### Are there cohomology classes on a hyperkähler manifolds which pull back to the Stiefel-Whitney classes on every Lagrangian submanifold?

This is a bit of a stab in the dark but I was wondering if anyone has defined cohomology classes on a hyperkähler manifold which pull back to the Stiefel-Whitney classes on any submanifold which is ...

**5**

votes

**0**answers

220 views

### About a non-degeneracy of Hodge-Riemann form..

Let $(M^{2n},\omega)$ be a closed connected symplectic manifold and let
$HR : H^2(M;R) \times H^2(M;R) \rightarrow R$ be the Hodge-Riemann form defined by
$HR(\alpha,\beta) = \int_M \alpha \beta ...

**5**

votes

**0**answers

394 views

### symplectic matrices

If $( A B | C D)$ is a symplectic matrix with entries in the finite field with two elements, is it necessarily the case that
$\sum_{i,j,k} a_{ij}b_{ij}c_{ik}d_{ik} = 0$?
This arose in connection ...

**5**

votes

**0**answers

441 views

### When should a moment polytope have “smooth” faces?

A codimension $d$ face of a polytope is called rationally smooth if it lies on only $d$ facets, because this is exactly the condition for the corresponding toric variety to have only orbifold ...

**4**

votes

**0**answers

86 views

### Gromov width of cotangent disk bundle

Given a symplectic manifold $(M^{2n},\omega)$, the Gromov width of $M$ is defined to be
$w(M)=sup\{{\pi r^2| B^{2n}(r) \rightarrow M}\}$
My question is: what is the explicit value of $w(D^*S^n)$, ...

**4**

votes

**0**answers

92 views

### Existence of a torus fibration with given vanishing cycles

Suppose I have a torus fibration over the disc with $n$ nodal singular fibers $F_1,\dots,F_n$ over the points $p_1,\dots,p_n$. I was specifically thinking about a Lagrangian fibration, but I'd be ...

**4**

votes

**0**answers

149 views

### Lagrangian fibration on Schoen's Calabi-Yau 3-fold

Schoen's Calabi-Yau 3-fold is the fiber product $X=Y_1\times_{\mathbb{P}^1}Y_2$ of two rational elliptic surfaces $Y_1\rightarrow\mathbb{P}^1$ and $Y_2\rightarrow\mathbb{P}^1$ with $\chi(X)=0$ and ...

**4**

votes

**0**answers

105 views

### Symplectic sum and Symplectic cut

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...

**4**

votes

**0**answers

124 views

### What can be said about compact embedded exact Lagrangians in the generalized pair of pants?

What can be said about compact embedded exact Lagrangians in the $n$-dimensional generalized pair of pants e.g. the hypersurface in $(\mathbb{C}^*)^{n+1}$ defined by the equation:
$$ 1+\Sigma_i z_i = ...

**4**

votes

**0**answers

100 views

### Concrete almost-complex structures on $3 \#CP^2$

The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex ...

**4**

votes

**0**answers

123 views

### Classification of Compact Symplectic Homogeneous Spaces

Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...

**4**

votes

**0**answers

392 views

### A generalization of Liouville formula for the determinant of a system of ODE?

Let $\Phi(t)$ be an $n\times n$ complex matrix whose columns are (independent) solutions of the system of ordinary differential equations (ODE):
$\frac{d}{dt}y= A(t) y$, where $A(t)$ is a ...

**4**

votes

**0**answers

233 views

### Lie-infinity structure in Lagrangian Floer theory ?

Is there (besides the A-infinity structure) also a L-infinity structure in Lagrangian Floer theory (forming together a G-infinity structure) - like in Hochschild cohomology ?

**4**

votes

**0**answers

421 views

### Almost-Kahler Einstein four manifolds

Are the odd Betti numbers of an Almost-Kahler Einstein four manifolds necessarily even ?

**4**

votes

**0**answers

216 views

### semisimple orbifold quantum cohomology

Dubrovin conjecture says, roughly speaking, that the quantum cohomology of a variety $X$ is semisimple if and only if it is a good Fano [good means that there exits a full exceptional collection in ...

**3**

votes

**0**answers

183 views

### Why is geometric quantization (esp. Berezin-Toeplitz quantization) interesting for a symplectic geometer/topologist?

I know that many symplectic geometers are interested in quantization as well.
From what I understood, quantization isn't expected to be used as a tool to answer symplectic questions (as in ...

**3**

votes

**0**answers

83 views

### DGBV algebra of symplectic manifold

Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...

**3**

votes

**0**answers

115 views

### Moment map in the singular case

The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...

**3**

votes

**0**answers

213 views

### Transversality in Bourgeois Oancea's non-equivariant contact homology

In their paper, "AN EXACT SEQUENCE FOR CONTACT- AND SYMPLECTIC HOMOLOGY", Bourgeois and Oancea defined, whenever there is sufficient transversality, a "non-equivariant contact homology". Essentially, ...

**3**

votes

**0**answers

127 views

### A symplectic version of critical points

According to the interesting comment of Mohammad F Tehrani, I revise the question as follows:
Assume $n>2$. For what type of compact n dimensional manifolds $M$ we can say:
For every smooth ...

**3**

votes

**0**answers

210 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 ...

**3**

votes

**0**answers

162 views

### Examples of non-Kahler symplectic manifolds.

Hi.
I want to find an example of symplectic (smooth compact) manifold $(M,\omega)$ whose Betti numbers are not unimodal, i.e.
$b_i(M) \leq b_{i+2}(M)$ fails for some $i < n$. ($ \dim{M} = 2n$.)
...

**3**

votes

**0**answers

248 views

### Is there a difference between a Picard stack and a stack of symmetric monoidal categories?

It seems to me that Picard stacks as defined by Deligne and used in algebraic geometry are stacks of symmetric monoidal categories. Am I missing something? Is there a difference between the two ...

**3**

votes

**0**answers

162 views

### Is tensor product flat with respect to the Hitchin connection?

Let $M$ be a compact symplectic manifold equipped with a line bundle $\mathcal L$ with curvature $\omega$. Denote by $H^0(M,\mathcal L)$ the space of smooth global sections of $\mathcal L$ (this ...