The geometry of the space of 2D shapes and the Weil-Petersson metric
by Kushnarev, Sergey, Ph.D., BROWN UNIVERSITY, 2010, 159 pages; 3430191

Abstract:

This thesis investigates three related topics on the geometry of the space of planar shapes.

In the first part, we setup the EPDiff (Euler-Poincaré equation on the group of diffeomorphisms) and investigate its singular solutions, Teichons, analogues of solitons. A symplectic integration scheme has been implemented to solve a Hamiltonian problem that arises in this part.

In the second part, the formula for the sectional curvature of the Weil-Petersson metric has been derived. It has been shown that the formula coincides with Arnold’s formula in the case of the metric without a null space.

The third part concerns numerical implementation of optimization problem of finding geodesics by minimizing the Weil-Petersson energy. The comparison of methods is shown alongside with the examples of geodesics.

 
AdviserDavid Mumford
SchoolBROWN UNIVERSITY
SourceDAI/B 71-11, p. , Nov 2010
Source TypeDissertation
SubjectsApplied mathematics; Mathematics
Publication Number3430191
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