Content and philosophy. We understand Computational Topology as the development of algorithmic tools implementing topological concepts for use in the sciences and engineering. This is different from using the computer to study topological questions although there is the potential for a beneficial symbiosis between the two efforts. The history of Computational Topology is short. It grew out of Computational Geometry as researchers expanded into applications where significant topological issues arise. The two such areas discussed in this course are structural molecular biology and geometric modeling. Both have connections to industries of substantial economical size.
A primary goal in this course is to develop a broad picture in which algorithmic tools connect pure mathematics with scientific applications. Our utilitarian view is that the application should drive the mathematics, the algorithms and the software development.
Organization. A typical day during the two weeks course consists of two general lectures by the principal speakers in the morning, each one-and-a-half hours in duration. There will be a more specialized one hour topical lecture after lunch. The speakers will vary and we will occasionally have introductions to topic related software packages. In the later afternoon there will be a loosely organized two hour brain-storming session.
We recommend the following texts for background reading.
Carl Brandon John Tooze. Introduction to Protein Structure. Garland, 1991.
Mark de Berg, Otfried Schwarzkopf, Marc van Kreveld, and Mark Overmars. Computational Geometry. Algorithms and Applications. Springer-Verlag, 1997.
Herbert Edelsbrunner. Geometry and Topology for Mesh Generation. Cambridge Univ. Press, 2001.
Yukio Matsumoto. An Introduction to Morse Theory. AMS, 2002.
James Munkres. Elements of Algebraic Topology. Addison Wesley, 1984.
Robert Tarjan. Data Structures and Network Algorithms. SIAM, 1983.