MATH 657: Computational Homology (Fall 2005)

Instructor: Sarah Day

Meeting Time & Room

Textbook: Computational Homology, by T. Kaczynski, K. Mischaikow, and M. Mrozek, Applied Mathematical Sciences 157, Springer 2004.

In recent years, algebraic topology has emerged as a powerful tool for studying the structure of mathematical objects captured by large sets of numerical or physical data. Objects depicted in this way are naturally stored as cubical complexes, to which all of the traditional notions of algebraic topology, and in particular homology, apply. With recent advances in computing power, and the development of a computational homology software package called CHomP, direct computation of the algebraic topological structure of cubical complexes has enabled researchers to gain a better understanding of large, often complicated, data sets and the mathematical objects they describe.

In this course we present the mathematical theory behind homology, discuss efficient means of computation, and study applications including computer-assisted proofs in nonlinear dynamics and measurement of the global structure of patterns. This course is aimed at a broad audience including (hopefully) students from mathematics, engineering, computer science, physics, and other related sciences. The prerequisites are minimal: a solid understanding of linear algebra and continuous functions. The students will undertake independent projects related to the material presented in the course, but tailored to their personal interests. Possible projects include topics in the applications previously mentioned as well as in image processing and the efficient implementation of algorithms for large data sets.

Last modified: April 1, 2005