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I'm trying to read the paper https://arxiv.org/pdf/2405.10732 to get a sense of things in more detail (probably quite helplessly).

It's talking about ellipticity. Should I be imagining a kind of tightly packed set of ellipses at all scales and shapes (kind of undulating or expanding and collapsing I guess)?

Does anyone have a better gloss-level sense of "the new method"?

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I'd guess the term "elliptic" has more to do with elliptic operators[1] than with ellipses. Of course, ultimately elliptic operators are named after ellipses, but the association is not as direct as you might imagine[2].

[1] https://en.wikipedia.org/wiki/Elliptic_operator

[2] https://mathoverflow.net/a/359723/111897

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