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Hamiltonian systems, symplectic flows, classical integrable systems

11 votes
1 answer
832 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 …
Nathaniel Bottman's user avatar
2 votes
Accepted

floer homology and viterbo's theorem

I think that the Floer complex decomposes as a direct sum of over the conjugacy classes of $\pi_1(M)$, no? Since the manifold on which we are doing infinite-dimensional Morse theory is of unbased, no …
Nathaniel Bottman's user avatar
11 votes

When do you go hunting for Lagrangian submanifolds?

One answer to your question "why one would be interested in Lagrangian submanifolds" is to quote Weinstein's Symplectic Creed (from a 1981 article), which says everything is Lagrangian --- i.e. (1) mo …
Nathaniel Bottman's user avatar
2 votes
Accepted

If a (linear) relation maps Lagrangian subspaces to Lagrangian subspaces, is it a Lagrangian...

Counterexample: set $U$ to be anything, $V := \text{pt}$, and $\Lambda \subset \overline{U} \oplus \text{pt}$ to be a non-Lagrangian subspace. (But maybe true with some hypotheses, e.g. $\Lambda$ ind …
Nathaniel Bottman's user avatar
3 votes

Square root for Hamiltonian diffeomorphisms

In a short paper posted last week, Peter Albers and Urs Frauenfelder prove that if $(M,\omega)$ is any closed symplectic manifold, then in any $\mathcal{C}^\infty$-neighborhood of the identity in $\te …
Nathaniel Bottman's user avatar
11 votes
1 answer
2k views

What is known about the strong Arnold conjecture?

Here are the two versions of Arnold's conjecture on Hamiltonian orbits: Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a nondegenerate Hamil …
Nathaniel Bottman's user avatar
8 votes
1 answer
410 views

How many "elementary" characterizations of twisted SU(2) representation varieties are known?

If $\Sigma_g$ is a genus-$g$ surface, $g \geq 2$, then let $\mathcal{M}(\Sigma_g)$ be its twisted SU(2) representation variety, i.e. $$\mathcal{M}(\Sigma_g) := \{ (A_1, B_1, \ldots, A_g, B_g) \in SU(2 …
Nathaniel Bottman's user avatar