A primer on topological persistence

The idea of topological persistence is to look at homological features that persist along a nested sequence of topological spaces. As a typical example, we may take the sequence of sublevel sets of a function. The combinatorial characterization of persistence in terms of pairs of critical values and fast algorithms computing these pairs make this idea practical and useful in dealing with the pervasive phenomenon of noise in geometric and visual data. This talk will 1. recall the relatively short history of persistence and some of its older roots; 2. introduce the concept intuitively while pointing out where algebra is needed to solidify the more difficult steps; 3. discuss a few applications to give a feeling of the potential of the method in dealing with noise and scale. Besides the initial concept, the talk will touch upon recent extensions and their motivation. © The Eurographics Association and Blackwell Publishing 2006.