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Transformations involving singular points

  1. Nov 6, 2008 #1
    Can a triangle be smoothly transformed to a circle?
  2. jcsd
  3. Nov 16, 2008 #2
    Either this question is too easy, too dumb or too hard. I'd just like an answer. I've found two two possibly relevant theorems:

    Hironaka: Every analytic space Y admits a resolution of singularities: there is a smooth manifold X and a proper map f:X->Y such that f is an isomorphism except over singular points.

    Castelnuovo, Enriques: Every singular surface has a unique minimal resolution of singularities.

    What exactly is a "minimal resolution"? It suggests to me that the singular points of a triangle cannot be "eliminated" by a smooth transformation, but only arbitrarily "minimized". Is this correct?
    Last edited: Nov 16, 2008
  4. Nov 20, 2008 #3
    People are reading, but not answering. I'll give a very simple special case example of how I think a singular point might be eliminated, not "minimally resolved" by means of a topological transformation:

    Consider an equalateral triangle circumscribed by a circle such that the three singular points lie on the circle. If the triangle is transformed to coincide with the circle (each line segment is transformed into a 2pi/3 radian arc), the singular points are eliminated.

    Can this idea be applied to the more general case of functions that are analytic almost everywhere, except for singular points?
    Last edited: Nov 20, 2008
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