
## 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.