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3. Invariants

From a differential geometry perspective we can assign invariants to a smooth curve or surface. This allows us to assign a signature curve to each curve.


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      Problem 3.1.

      How complete are these invariants? If two curves have the same signature what can we say about them?


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          Problem 3.2.

          What can we say about the degree of a signature curve? How is it related to the automorphism group of the original curve?


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              Problem 3.3.

              What is the signature curve of a general canonical curve?


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                  Problem 3.4.

                  What can we say about the signature of the signature of a curve?


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                      Problem 3.5.

                      Which curves occur as signatures of another curve?


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                          Problem 3.6.

                          Can we use differential (integral, other…) invariants to construct a moduli space?


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                              Problem 3.7.

                              How can we use these signatures for object recognition?


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                                  Problem 3.8.

                                  Find local invariants for a natural (Lie) groupoid action. (e.g. bathroom tiles in an irregular bathroom). How is this related to quasi-invariants?

                                      Cite this as: AimPL: Algebraic vision, available at http://aimpl.org/algvision.