On data, information theory and topology.
I would like to show you how tools from computational topology can be used for analyzing various kinds of data.
First, we develop topological intuition starting from low dimensional data such as 2-dimensional images coming from astrophysics, 3-dimensional medical scans of bones, and atomic configurations of amorphous solids.
Then we move to a more abstract setting: high dimensional point-cloud data representing collections of objects, such as images or text documents. In many situations, such data is best measured with distances coming from information theory. These distances often lack symmetry and violate the triangle-inequality, whereas standard topological methods were developed for the well-behaved Euclidean distance. Still, I will try to convince you — with the help of some colorful 4D printouts — that extending topological methods to this non-metric setting not only gives rise to beautiful mathematics, but also makes practical sense.