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Mathematics Colloquium

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Date:  Tuesday, November 07, 2006

Title:  The geometry of the space of simple closed plane curves

Abstract:  The space of simple closed plane curves is the simplest case of a "Differential-Geometric Chow variety", the space of submanifolds of some type in a fixed ambient manifold. These are infinite dimensional manifolds, actually not differentiable Hilbert manifolds but topological Hilbert manifolds -- which shows how fundamentally nonlinear they are. A big surprise is that they carry a wealth of quite different Riemannian metrics, whose geodesics and curvature have very distinct properties. I will sketch the situation but focus on the Weil-Peterson metric which works only for 2D ambient spaces. In both the 2D and 3D cases, the theory has applications to computer vision, e.g. to universal atlases of the human body.


Speaker:  David Mumford
Institution:  Brown University

 

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