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Ph.D. Theses

Algorithms and Hardness Results in Computational Homology

By Chao Chen
Advisor: Daniel Freedman
July 9, 2009

The need to identify and quantify topological features, such as homology classes over the Z_2 field, has become apparent in a variety of academic fields. In this thesis, we present algorithms and hardness results concerning the localization and measurement of homology classes, as well as the computation of natural generators of the homology group. The idea is to give each class a measure using the geometry of the underlying space, and to represent each class with a geometrically concise cycle. We also discuss an open problem of computing the structural relationship between persistent homology classes, and finally, an application of computational topology in a biological problem, ribosome drug docking.

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