Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

Filter by
Sorted by
Tagged with
100 votes
5 answers
11k views

Is there a high level reason why the inverse square law of gravitation yields periodic orbits without precession?

Given a spherically symmetric potential $V: {\bf R}^d \to {\bf R}$, smooth away from the origin, one can consider the Newtonian equations of motion $$ \frac{d^2}{dt^2} x = - (\nabla V)(x)$$ for a ...
Terry Tao's user avatar
  • 108k
75 votes
10 answers
17k views

What is a Lagrangian submanifold intuitively?

What are good ways to think about Lagrangian submanifolds? Why should one care about them? More generally: same questions about (co)isotropic ones. Answers from a classical mechanics point of view ...
Jan Weidner's user avatar
  • 12.8k
70 votes
10 answers
11k views

The Planck constant for mathematicians

The questions Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant? Q2. How does the Planck constant appear in mathematics of quantum mechanics? In ...
Gil Kalai's user avatar
  • 24.2k
67 votes
4 answers
9k views

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 ...
John Pardon's user avatar
  • 18.3k
56 votes
9 answers
7k views

Examples in mirror symmetry that can be understood.

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider ...
aglearner's user avatar
  • 14k
53 votes
0 answers
2k views

What is the current status of derived differential geometry?

I hope you will excuse this naive and general question. I've read from many places (e.g. Dominic Joyce's website, John Pardon's thesis, etc.) that the/a "right" foundations for many ...
davik's user avatar
  • 2,035
50 votes
4 answers
16k views

What is a symplectic form intuitively?

Hi, to completely describe a classical mechanical system, you need to do three things: -Specify a manifold $X$, the phase space. Intuitively this is the space of all possible states of your system. -...
Jan Weidner's user avatar
  • 12.8k
49 votes
5 answers
4k views

are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
symplectomorphic's user avatar
43 votes
0 answers
2k 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 ...
ARupinski's user avatar
  • 5,181
42 votes
2 answers
4k views

About a letter by Richard Palais of 1965.

In Cushman and Bates, Global Aspects of Classical Integrable Systems, 1997, I have read In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of ...
agt's user avatar
  • 4,246
42 votes
1 answer
3k views

Is a symplectic camel actually prohibited from passing through the eye of a needle?

Gromov's symplectic nonsqueezing theorem asserts that in the symplectic space ${\bf R}^{2n}$ with canonical coordinates $p_1,\dots,p_n,q_1,\dots,q_n$, and two radii $0 < r < R$, it is not ...
Terry Tao's user avatar
  • 108k
41 votes
5 answers
4k views

Can cotangent bundles see exotic smooth structures?

I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school. Caveat: I'm not a ...
Marco Golla's user avatar
  • 10.4k
39 votes
3 answers
5k views

What is so geometric about symplectic geometry?

Symplectic geometry is often motivated by the Hamilton's equation which in turn are a reformulation of Newton's third law. But the subject itself is of independent mathematical interest. What I don't ...
Jannik Pitt's user avatar
  • 1,103
35 votes
4 answers
6k views

When is a symplectic manifold equivalent to a cotangent bundle?

Let $X$ be a differentiable manifold. Its cotangent bundle $T^*X$ carries a canonical 1-form $ \alpha$ whose exterior differential $\omega = d\alpha$ endows $T^*X$ with the structure of a symplectic ...
Per Vognsen's user avatar
  • 2,041
35 votes
1 answer
1k views

What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ (h^{...
Lev Borisov's user avatar
  • 5,156
34 votes
6 answers
5k views

Has anything precise been written about the Fukaya category and Lagrangian skeletons?

At some point in this past year, some Fukaya people I know got very excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons." As I understand it, a Lagrangian ...
Ben Webster's user avatar
  • 43.9k
34 votes
6 answers
10k views

Roadmap for Mirror Symmetry

I am interested in learning Mirror Symmetry, both from the SYZ and Homological point of view. I am taking a reading course in Mirror Symmetry, which will focus on the SYZ side. I know basic Complex ...
J Verma's user avatar
  • 3,178
33 votes
5 answers
8k views

How to see the Phase Space of a Physical System as the Cotangent Bundle

Two things today motivated this question. First, the professor said that in a lecture Thurston mentioned Any manifold can be seen as the configuration space of some physical system. Clearly we ...
B. Bischof's user avatar
  • 4,782
29 votes
1 answer
2k views

High-Dimensional Analogs of Polygon Spaces

[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.] Background: Polygon spaces Given a ...
Gil Kalai's user avatar
  • 24.2k
28 votes
9 answers
12k views

Book on symplectic geometry

Can someone please tell me some introductory book on symplectic geometry? I have no prior idea of the subject but I do know about Lagrangian and Hamiltonian dynamics (at the level of Landau-Lifshitz ...
Debangshu Mukherjee's user avatar
28 votes
6 answers
3k views

Why is there no symplectic version of spectral geometry?

First, recall that on a Riemannian manifold $(M,g)$ the Laplace-Beltrami operator $\Delta_g:C^\infty(M)\to C^\infty(M)$ is defined as $$ \Delta_g=\mathrm{div}_g\circ\mathrm{grad}_g, $$ where the ...
B K's user avatar
  • 1,890
28 votes
4 answers
3k views

When are two symplectic forms "isotopic"?

I've been skulking around MathOverflow for about a month, reading questions and answers and comments, and I guess it's about time I asked a question myself, so here is one has interested me for a long ...
Dick Palais's user avatar
  • 15.1k
27 votes
2 answers
1k views

Intuition for symplectic groups

My question essentially breaks down to How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
Robin Goodfellow's user avatar
26 votes
5 answers
10k views

What does the word "symplectic" mean?

I know the definition of symplectic structure, symplectic group, and so on. But what does the word "symplectic" itself mean? Meta question: I have many other mathematical words whose etymologies are ...
Yuji Tachikawa's user avatar
26 votes
2 answers
2k views

Manifolds distinguished by Gromov-Witten invariants?

What is a simplest example of a manifold $M^{2n}$ that admits two symplectic structures with isotopic almost complex structures, and such that Gromov-Witten invariants of these symplectic structures ...
Dmitri Panov's user avatar
  • 28.7k
25 votes
4 answers
7k views

Is the Fukaya category "defined"?

Sometimes people say that the Fukaya category is "not yet defined" in general. What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact ...
Kevin H. Lin's user avatar
  • 20.7k
25 votes
1 answer
1k views

What can we say about the Cartesian product of a manifold with its exotic copy?

Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$. Is it true that $M\times M$ is diffeomorphic to $M\times M^E$? I am ...
Anubhav Mukherjee's user avatar
25 votes
2 answers
2k 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 ...
Rogier Swierstra's user avatar
24 votes
3 answers
3k views

Classical mechanics motivation for poisson manifolds?

Suppose I want to understand classical mechanics. Why should I be interested in arbitrary poisson manifolds and not just in symplectic ones? What are examples of systems best described by non ...
Jan Weidner's user avatar
  • 12.8k
24 votes
4 answers
5k 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, ...
nicolas's user avatar
  • 683
24 votes
1 answer
2k views

Why are Lagrangian submanifolds called Lagrangian?

Much of the terminology in symplectic geometry comes from classical mechanics: the symplectic manifold is modeled on a cotangent bundle $T^*N$ of some configuration space $N$ with local position ...
dorebell's user avatar
  • 2,968
23 votes
7 answers
9k views

Introduction to Floer Theory?

Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive ...
23 votes
3 answers
1k views

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 ...
Dmitri Panov's user avatar
  • 28.7k
22 votes
5 answers
7k views

The Jacobi Identity for the Poisson Bracket

It is well known that if $M, \Omega$ is a symplectic manifold then the Poisson bracket gives $C^\infty(M)$ the structure of a Lie algebra. The only way I have seen this proven is via a calculation in ...
Paul Siegel's user avatar
  • 28.7k
22 votes
4 answers
2k views

What is an "integrable hierarchy"? (to a mathematician)

This is one of those "what is an $X$?" questions so let me apologize in advance. By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so ...
Saal Hardali's user avatar
  • 7,549
22 votes
2 answers
3k views

What is the relationship between integrable systems and toric degenerations?

Given an integrable system on a Kahler manifold X, is there a way to associate a toric degeneration of X whose Milnor fibers are related to the fibers of the integrable system? An integrable system ...
David Treumann's user avatar
22 votes
1 answer
868 views

Pseudo-holomorphic curves in the six-sphere

Equip $S^6$ with the almost complex structure coming from the cross product on $\mathbb R^7$ (i.e. the product on the pure imaginary octonions). What is known about the psudo-holomorphic curves in ...
John Pardon's user avatar
  • 18.3k
22 votes
5 answers
2k views

When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold?

Here is my precise question. Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function. The symplectic form gives rise to an isomorphism between the tangent ...
Paul Siegel's user avatar
  • 28.7k
22 votes
2 answers
683 views

$(M,\omega)$ not symplectomorphic to $(M,-\omega)$

Looking for an example of a symplectic manifold $(M,\omega)$ that is not symplectomorphic to $(M,-\omega)$. In particular this means that $M$ must be chiral (i.e. doesn't admit an orientation-...
Emily Maw's user avatar
  • 697
21 votes
2 answers
2k views

A royal road to Coulomb branches of 3D $\mathcal{N}=4$ gauge theories

So, I've been very interested recently with the developements of the (now mathematically precise) theory of Coulomb branches - in particular because of its recent applications on representation theory ...
jg1896's user avatar
  • 2,683
21 votes
3 answers
2k views

Proof of Giroux's correspondence

It is extensively used and cited the following statement due to Giroux: Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and ...
Paul's user avatar
  • 1,379
21 votes
2 answers
2k views

Infinite dimensional symplectic geometry

Could anyone comment on possible references concerning infinite dimensionsal symplectic manifolds?. I am mainly concerned with hilbert spaces, so i am not interested in the convenient analysis ...
Javier's user avatar
  • 483
21 votes
3 answers
2k views

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 ...
Yuji Tachikawa's user avatar
21 votes
1 answer
1k views

Why symplectic geometry gives Poisson geometry

One way I've learned to understand Poisson geometry is to consider it as symplectic geometry with no open conditions - i.e. no condition of nondegeneracy. This idea can be applied to many other ...
user44191's user avatar
  • 4,961
20 votes
4 answers
3k views

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 ...
agt's user avatar
  • 4,246
20 votes
3 answers
3k views

Why are Gromov-Witten invariants of K3 surfaces trivial?

Why is GW invariants of K3 surfaces are trivial? My naive guess is that GW invariants are deformation invariant and you can always deform your K3 surface to non-projective one, which has no subcomplex ...
Daniel's user avatar
  • 231
20 votes
2 answers
3k views

What is the significance that the Springer resolution is a moment map?

Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution $$ \mu: T^*\mathcal{B}\rightarrow \mathcal{N} $$ is the moment ...
Zhaoting Wei's user avatar
  • 8,657
20 votes
1 answer
2k views

Functoriality of the cotangent bundle

Recall that to any manifold $X$, I can assign in a canonical way a manifold $\mathrm T ^* X$, the total space of the cotangent bundle over $X$. Recall also that, unlike the tangent bundle ...
Theo Johnson-Freyd's user avatar
19 votes
3 answers
3k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
user avatar
19 votes
3 answers
939 views

How big can a wedge of $n$ 2-forms in $\mathbb{R}^{2n}$ be?

$\def\RR{\mathbb{R}}$Let $\omega$ be a $2$-form on $\RR^{2n}$, where $\RR^{2n}$ has the usual Euclidean norm. The comass of $\omega$ is defined to be $\max_{|u|, |v| \leq 1} \omega(u,v)$. Here is the ...
Mikhail Katz's user avatar
  • 15.1k

1
2 3 4 5
29