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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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How many circular cylinders through 5 general points?

If the 5 points are generic then I believe Lemma 7.1 in arXiv: 1306.2346 is relevant to your question. The Lemma, for the case of cylinders, says if there are two cylinders that pass through the same …
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Examples of naturally occurring Quadratic forms or quadrics.

Quadratic forms occur naturally, as stress energy functions, in Bob Connelly's work on tensegrity structures. You can find a wealth of details on his website, in particular the book http://www.math.c …