Date of Award

2006

Degree Name

Mathematics

College

College of Science

Type of Degree

M.A.

Document Type

Thesis

First Advisor

Peter Saveliev

Second Advisor

Ralph Oberste-Vorth

Third Advisor

Scott Sarra

Fourth Advisor

Philippe Georgel

Fifth Advisor

Jo Fuller

Abstract

Homology is a field of topology that classifies objects based on the number of n- dimensional holes (cuts, tunnels, voids, etc.) they possess. The number of its real life ap- plications is quickly growing, which requires development of modern computational meth- ods. In my thesis, I will present methods of calculation, algorithms, and implementations of simplicial homology, alpha shapes, and persistent homology.

The Alpha Shapes method represents a point cloud as the union of balls centered at each point, and based on these balls, a complex can be built and homology computed. If the balls are allowed to grow, one can compute the persistent homology, which gives a better understanding of the shape of the object represented by the point cloud by eliminating noise.

These methods are particularly well suited for studying biological molecules. I will test the hypothesis that persistent homology can describe some important features of a protein's shape.

Subject(s)

Homology theory.

Topology.

Triangulation.

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