Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,427
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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 ...
-3
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1
answer
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Quantum dynamics on varieties and Salmon Prizes
Concluding Progressive Remarks
A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize.
The Salmon Prize (photo of the ...
5
votes
2
answers
593
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contactomorphism of $S^{2n+1}$ for n>1
Is there some result about contactomorphism groups of $S^{2n+1}$ or $T^{2n-1}$ for n>1?
For example, do we know the rank of $\pi_{i}(Cont(S^{2n+1})) \otimes \mathbb{Q}?$ where "Cont" means the ...
7
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2
answers
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Oriented double normals
Given an embedded two-torus in three-dimensional Euclidean space, paint the inside of the torus red and the outside blue. Show that there is an oriented line in ${\mathbb R}^3$ that cuts the torus ...
1
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0
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Co-normal bundle of orthogonal compliment
Is the following fact well known?
Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \...
10
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4
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References: Infinite dimensional Lie algebras
What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras...) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; ...
0
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1
answer
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intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$
Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. $\omega(\...
67
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4
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Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?
The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
17
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4
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Can you tell the volume of a symplectic manifold from the Poisson brackets?
Suppose $(X^{2n},\omega)$ is a compact symplectic manifold. Knowing the algebra $C^\infty(X)$ is equivalent to knowing the manifold $X$, and knowing the Poisson bracket $\{\cdot,\cdot\}:C^\infty(X)\...
2
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1
answer
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density of lagrangian grassmannian in usual grassmannian.
Consider the canonical symplectic structure $(\omega, J)$ on $\mathbb{R}^{2n}$.
(i) What can be said about the density of the lagrangian grassmannian $L$ (i.e. those rank $n$ totally isotropic linear ...
19
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2
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Can a Lagrangian submanifold of ${\mathbb R}^{2n}$ be dense ($n>1$)?
I'm betting `yes, sure!', but don't see it. Could someone please point me toward,
or construct for me, a Lagrangian submanifold immersed in
standard symplectic ${\mathbb R}^{2n}$ for $n > 1$, ...
8
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2
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2k
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Places to learn about Landau-Ginzburg models
Here is what I know about Landau-Ginzburg models:
It is an important player in the story of mirror symmetry.
It involves "potentials" which are functions of complex varibles, which have isolated ...
10
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0
answers
797
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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 ...
21
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When is a coadjoint orbit an integrable system (in a weak sense explained below)?
Let $X$ be an affine holomorphic symplectic variety of dimension $2n$, with the associated Poisson bracket { , }. Let's say it's an integrable system when there are $n$ algebraically independent ...
1
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2
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reduced symplectic form
If you take a $G$ as any group acting on the dual of its lie algebra $g^{*}$ by the coadjoint action it is known that the coadjoint orbit $O_{\alpha}$ is a symplectic manifold with the KKS symplectic ...
9
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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 long!...
4
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1
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673
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Does symplectic cohomology admit a natural Batalin-Vilkovisky algebra structure?
Symplectic cohomology is the closed-string sector of the open-closed TCFT associated to the A-side topologically twisted theory.
For general reasons, it should carry the structure of an algebra over ...
2
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0
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In search for a more geometric proof of a result of van der Schaft and Maschke on nonholonomic mechanics
Edit: Now I have found something that appears to answer my own question. It is section 2 in the paper "On Submanifolds and Quotients of Poisson and Jacobi Manifolds" by Ch.-M. Marle. (There, he ...
6
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1
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423
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Lagrangian submanifold containing a curve
Suppose $(M,\omega)$ is a compact symplectic manifold and $C$ a closed curve in it. Is there a Lagrangian submanifold containing $C$?
I have a sequence of $J_i$-holomorphic maps from a disk to $M$, ...
7
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1
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433
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Test for Homogeneity of Symplectic Manifolds?
How can one tell whether a given (finite-dimensional) symplectic manifold is homogeneous (that is, admits a transitive group of symplectomorphisms)?
Note that it is of little help to observe that a ...
0
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2
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1k
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Hodge decomposition of a symplectic form.
Can anyone explain to me what the Hodge decomposition form of a symplectic form in a special symplectic manifold looks like?
3
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1
answer
391
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Hamiltonian Reduction and Affine quotient
Given a smooth affine symplectic variety $V$ with an action of a connected algebraic $G$. If $\mu$ is the moment map, the define the affine quotient to be :
$X = \mu^{-1}(0)// G = \text{Spec}\mathbb{...
2
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0
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199
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Non-regular (Non-coherent) subdivisions of a polygon.
There are many papers and books which study about the regular subdivision of a convex lattice polytope.
My question is about "Non"-regular subdivisions of a 2-dimensional convex lattice polygon.
I ...
6
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0
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653
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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 ...
4
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3
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515
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Obstructions to being a hyperplane section or a fibre of a Lefschetz pencil
Given a smooth projective variety $X$, when could $X$ fail to be a hyperplane section in some other variety $Y$, or fail to be the fibre of some Lefschetz pencil $\widetilde{Y} \rightarrow \mathbb{P}^{...
3
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2
answers
783
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Dimension of moduli space in Lagrangian Floer homology
Let $(M,\omega)$ be symplectic manifold with $\omega=c_{1}=0$ on
$\pi_{2}M$. Let $\Lambda\subseteq M$ be Lagrangian submanifold.
Let $H:M\times S^{1}\rightarrow\mathbf{R}$ be Hamiltonian and $J$
be ...
3
votes
1
answer
645
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Elliptic estimates and regularity of the $\overline{\partial}$-operator with totally real boundary conditions in $W^{1,p},$ $1<p\le 2$
Let $\Omega$ be an open subset of the upper half-plane in the complex plane. I am considering the following problem:
(1) $\overline{\partial}u=f,$ $\textrm{Im} f=0$ on the real line for maps complex-...
2
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0
answers
349
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Dissipative Hamiltonian System with a Periodic Force
Let $H:P \to \mathbb{R}$ be a Hamiltonian on a symplectic manifold $(\omega,P)$ and let $X_H: P \to TP$ be the Hamiltonian vector-field. Let $F:P \to T^*P$ be a dissipative force field such that for $...
6
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3
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793
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Kähler structure on a complex reductive group
Let $G$ be a complex reductive group, and $K$ a maximal compact subgroup (such that $K_{\mathbb{C}}=G$). By the polar decomposition theorem one has that, as manifolds, $G\cong T^*K$. The inherited ...
9
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1
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577
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Classical analogue of the Stone-von Neumann Theorem?
Let $U_s$, $V_t$ be a pair of continuous $n$-parameter groups ($n < \infty$) of unitary operators on a complex Hilbert space $\mathcal{H}$. The Stone-von Neumann Theorem establishes that any such ...
6
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1
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453
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Splitting principle in equivariant cohomology
The following is a weaker version of what is called splitting principle in
Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6:
Let $G$ be a compact ...
1
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0
answers
222
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(Non-)Surjectivity of the Maslov index
Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over $...
0
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1
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588
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Pullback of D-Modules
Hi,
just a short question on the theory of D-Modules:
if one has $p:X\times Y \rightarrow Y$ the projection of two smooth projective complex varieties to the second factor, then what is a $p^{*}D_Y$-...
6
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3
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1k
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Question about the definition of hamiltonian group action.
So I'm reading the part in Ana Cannas da Silva's book "Lectures on Symplectic Geometry" available (on her website) about hamiltonian group actions on a symplectic manifold. She starts by defining $\...
3
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3
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412
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Open symplectic embeddings and deformation quantization
I would like to know whether there is a specific relationship between the deformation quantizations of the Poisson algebras of the functions on a symplectic manifold, say $M$, and of the functions on ...
2
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1
answer
456
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Non-Abelian Duistermaat-Heckman Measure (not just a reference request)
Let $V$ be a representation of some torus $T$. It is then well-known that the Duistermaat-Heckman measure for $P(V)$ is the weak limit of the properly rescaled distribution of multiplicities of ...
2
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1
answer
661
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How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?
What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...
3
votes
1
answer
519
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How many Symplectic Structures on CP^n?
Is there any classification result(s) regarding how many symplectic structures on CP^n?
4
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0
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210
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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 ...
6
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1
answer
842
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How can I see the "missing" Poisson center when the rank of the Poisson structure drops?
Recall that a Poisson algebra is a commutative algebra $A$ along with a bracket $\lbrace,\rbrace: A^{\otimes 2} \to A$ which is a Lie bracket and which is also a derivation in each variable. The ...
17
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1
answer
888
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Finite fundamental groups of 3-dimensional Calabi-Yau manifolds
Question. Is there an example of a compact $3$-dimensional Calabi-Yau manifold with finite fundamental group $G$ that does not admit a free action on $S^3$?
This question is motivated by the ...
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0
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615
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About automorphisms of ratonal surfaces.
Hi. I have a question about automorphisms of smooth ratonal surfaces. (I am not an algebraic geometer, so my question could be stupid. I am sorry about that.)
Let $X_k$ be a blow-up of $\mathbb{P}^2$ ...
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4
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What is the role of contact geometry in the hamiltonian mechanics?
Let us assume someone is interested in the study of Hamiltonian mechanics.
What are good examples to illustrate him of the usefulness of contact geometry in this context?
On one hand the Hamiltonian ...
4
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0
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Do real polarization and Kahler polarization of character varieties of closed surfaces give equivalent representations of the Mapping Class Group?
This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$.
These ...
15
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1
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Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?
Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper.
Heegaard ...
4
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3
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Example of symplectic and hamiltonian diffeomorphism on $S^2$ and $T^2$
Hello everybody. For a purpose of consolidation of some result I am trying to set down, I need to construct an example to sustain the theory and I am looking for symplectic and Hamiltonian ...
24
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3
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Hyperbolic Coxeter polytopes and Del-Pezzo surfaces
Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf
I would like to find a reference for a beautiful ...
3
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1
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Example in Guillemin-Sternberg's Convexity Paper
At the end of the introduction of Guillemin-Sternberg's 1982 Inventiones paper "Convexity Properties of the Moment Mapping", there is a nice picture of a non-Abelian moment polytope attributed to ...
4
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0
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425
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$A_{\infty}$-categories and mirror symmetry
I have been looking for an electronic version of Kontsevchi's talk in the Arbeitstagung at 1993. In Kontsevich's publications webpage, I found this reference preprint MPI/93-57, but there is no link ...
1
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Mirror Symmetry and Quantum Gravity [closed]
Can we use the mirror symmetry to define quantum gravity ?
It may be fair to say that so far we don't know how to quantize a Riemannian manifold
(or a complex manifold). But a symplectic manifold ...