| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

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.
    1. Problem 3.1.

      How complete are these invariants? If two curves have the same signature what can we say about them?
        • 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?
            • Problem 3.3.

              What is the signature curve of a general canonical curve?
                • Problem 3.4.

                  What can we say about the signature of the signature of a curve?
                    • Problem 3.5.

                      Which curves occur as signatures of another curve?
                        • Problem 3.6.

                          Can we use differential (integral, other…) invariants to construct a moduli space?
                            • Problem 3.7.

                              How can we use these signatures for object recognition?
                                • 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.