Solved?

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

Badly written

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The conjecture is described in the article as follows:

"Let   be a compact symplectic manifold. For any smooth function  , the symplectic form   induces a Hamiltonian vector field   on  , defined by the identity

 

"The function   is called a Hamiltonian function.

"Suppose there is a 1-parameter family of Hamiltonian functions  , inducing a 1-parameter family of Hamiltonian vector fields   on  . The family of vector fields integrates to a 1-parameter family of diffeomorphisms  . Each individual   is a Hamiltonian diffeomorphism of  .

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

Merge proposal: merge Arnold–Givental conjecture into this article

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The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.
The result of this discussion was to merge (see WP:SILENCE). Mathwriter2718 (talk) 11:52, 21 June 2024 (UTC)Reply

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)Reply

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.