This talk is an introduction to applications of topology to (A) sensor networks and (B) data analysis.
(A) Imagine a collection of ball-shaped sensors are scattered in a bounded domain. Unfortunately the sensors can't measure their locations, and instead they only measure which sensors overlap. Can you use this connectivity data to determine if the sensors cover the entire domain? Topology provides tools for this coverage problem. Suppose now that the sensors are moving; an evasion path exists if a moving intruder can avoid overlapping with any sensor. Can you use the time-varying connectivity data of the sensor network to decide whether an evasion path exists?
(B) The shape of a dataset often reflects important patterns within. Two such datasets with interesting shapes are a space of 3x3 pixel patches from optical images, which can be well-modeled by a Klein bottle, and the conformation space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles of singularities. I will describe how computational topology provides tools for understanding high-dimensional datasets which are otherwise hard to visualize.