Sorry for the delayed response; I'm "submitting too fast".
> I did qualify my statement with "arguably"
I always tell my students—and, though it's a joke, I think that it's more true than casual consideration might suggest—that the correct answer (though not one that they're allowed to use on tests!) to any question in mathematics is "it depends." The more elementary the question, the more profound it sounds that you can see angles from which different answers can be correct. (Of course, backing this up with actual profound re-interpretations is the hard part. :-) )
> That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper (http://arxiv.org/pdf/1505.03840.pdf), which briefly mentions how one might extend the concept to non-compact groups, but doesn't elaborate further.
As a decidedly pure mathematician, that's outside my wheelhouse, although I did have the pleasure of having some of the applications explained to me by one of the authors of one of the papers on cryo-electron microscopy cited in the linked article. I'm afraid that the idea of embedding a non-compact group in a larger, compact group is foreign to my way of thinking about things, so (in case you were implicitly asking) unfortunately I can't shed any light there.
> I did qualify my statement with "arguably"
I always tell my students—and, though it's a joke, I think that it's more true than casual consideration might suggest—that the correct answer (though not one that they're allowed to use on tests!) to any question in mathematics is "it depends." The more elementary the question, the more profound it sounds that you can see angles from which different answers can be correct. (Of course, backing this up with actual profound re-interpretations is the hard part. :-) )
> That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper (http://arxiv.org/pdf/1505.03840.pdf), which briefly mentions how one might extend the concept to non-compact groups, but doesn't elaborate further.
As a decidedly pure mathematician, that's outside my wheelhouse, although I did have the pleasure of having some of the applications explained to me by one of the authors of one of the papers on cryo-electron microscopy cited in the linked article. I'm afraid that the idea of embedding a non-compact group in a larger, compact group is foreign to my way of thinking about things, so (in case you were implicitly asking) unfortunately I can't shed any light there.