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I have another short question regarding terminology. The phonetic similarity of simplex and symplectic manifold has little or nothing to do with any mathematical relationship, correct?

I always considered a symplectic manifold to be a smooth manifold with an associated two form and a simplex to be a generalized tetrahedron. In conversation, someone asked me if any relationship existed between them, and I responded that there was none of which I was aware. I wanted to see if anyone could offer a potential connection or validate the absence of one.

Thank you all.

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I've read that Weil came up with the word "symplectic" mimicking the already in use word "complex" of latin origin. Instead, he used the corresponding Greek roots: συν/συμ (syn/sym) = with/together and πλέκω (roughly to braid/weave) to create symplectic "meaning" woven together or roughly "complex."

On a slightly ironic mathematical note, there is in some sense a relationship between convex geometry and symplectic geometry. Given a Hamiltonian torus action by $T$ on a symplectic manifold $(M,\omega)$, there is an associated moment map $\Phi:M\to \mathfrak{t}^*$. The image of $M$ under $\Phi$ turns out to be a polytope in $\mathfrak{t}^*$. This result is due to Atiyah-Guillemin-Sternberg, I believe. This was not the motivation for the name, unless I'm mistaken.

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    $\begingroup$ There is also a beautiful connection between the Alexander-Fenchel inequalities for convex bodies and the Hodge inequalities for toric varieties in algebraic geometry. Unfortunately, I don't have a reference for this. $\endgroup$
    – Deane
    Apr 1 '21 at 21:26
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    $\begingroup$ That sounds beautiful indeed. If you remember a reference, it'd be great if you left it here :-). $\endgroup$ Apr 1 '21 at 21:50
  • $\begingroup$ Thank you for your excellent insight! If I may risk oversimplification, the names "symplectic" and "simplex" had no intended relationship originally; however, they acquired connections by later mathematical results. As an unrelated question, do you have any opinions on the Differential Geometry Volumes by M. Spivak and the book Differential Topology by J. Milnor? $\endgroup$
    – JPwin
    Apr 1 '21 at 22:27
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    $\begingroup$ I haven't read Spivak, but I think Milnor's book is quite good. $\endgroup$ Apr 1 '21 at 22:32
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    $\begingroup$ @JPwin, yes, I think that has at least some aspects of what I mean. $\endgroup$
    – Deane
    Apr 1 '21 at 22:34

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