Talk:Arnold conjecture

Solved?
Is this conjecture still open? Didn't Floer solve this? 77.3.23.230 (talk) 11:25, 31 July 2023 (UTC)

Badly written
The conjecture is described in the article as follows:

"Let $$(M, \omega)$$ be a compact symplectic manifold. For any smooth function $$H: M \to {\mathbb R}$$, the symplectic form $$\omega$$ induces a Hamiltonian vector field $$X_H$$ on $$M$$, defined by the identity

$$\omega( X_H, \cdot) = dH.$$

"The function $$H$$ is called a Hamiltonian function.

"Suppose there is a 1-parameter family of Hamiltonian functions $$H_t: M \to {\mathbb R}, 0 \leq t \leq 1$$, inducing a 1-parameter family of Hamiltonian vector fields $$X_{H_t}$$ on $$M$$. The family of vector fields integrates to a 1-parameter family of diffeomorphisms $$\varphi_t: M \to M$$. Each individual $$\varphi_t$$ is a Hamiltonian diffeomorphism of $$M$$.

"The Arnold conjecture says that for each Hamiltonian diffeomorphism of $$M$$, it possesses at least as many fixed points as a smooth function on $$M$$ possesses critical points."

The last sentence, which finally describes the actual conjecture, make no reference to anything that came before. Surely this can be written much more clearly so that the connection of the conjecture to what preceded it is clear.


 * I agree that this is not written so clearly. The connection to what came before is that the "before" defines Hamiltonian diffeomorphisms, which is used in the statement of the conjecture. Mathwriter2718 (talk) 11:41, 13 June 2024 (UTC)

Merge proposal: merge Arnold–Givental conjecture into this article
I propose merging Arnold–Givental conjecture into this article. The Arnold–Givental conjecture is a generalization of one of the versions of the Arnold conjecture. Indeed, if you look at Arnold–Givental conjecture page, you will see that all of the setup for the conjecture (which is half of that article) overlaps with the setup that is already in this article. Further, these articles are both pretty small. Mathwriter2718 (talk) 03:34, 13 June 2024 (UTC)