Dissertation/Thesis Abstract

The geometry of the space of 2D shapes and the Weil-Petersson metric
by Kushnarev, Sergey, Ph.D., Brown University, 2010, 159; 3430191
Abstract (Summary)

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.

Indexing (document details)
Advisor: Mumford, David
School: Brown University
School Location: United States -- Rhode Island
Source: DAI-B 71/11, Dissertation Abstracts International
Subjects: Applied Mathematics, Mathematics
Keywords: Planar shapes, Sectional curvature, Space, Weil-Petersson metric
Publication Number: 3430191
ISBN: 978-1-124-30206-5
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