Algebraic topology from loop spaces to the shape of data

In this talk I will present two branches of my research, thematically connected by the use of algebraic constructions to describe shape.

First, I will introduce loop spaces, which are a special kind of mapping space, and their role in string topology. I will outline my work on algebraic models for their homology and show how a family of polytopes arises naturally from this problem. These polytopes are interesting in their own right, and we will observe some of their unusual properties.

Next, we will turn to geometric patterns in data. Clustering is a familiar way to describe groups of nearby points. But what about points arranged around a circle, a sphere, or data that "has holes"? Persistent homology describes such structures across scales and dimensions, turning “shape” into algebra. As such, it is particularly well suited to studying pattern formation and self-organization in complex systems. We will see this method in action by looking at "how the zebra(fish) gets its stripes".

Lastly, I will discuss ongoing work towards a persistent homology approach to -omics analysis, potential applications in biology and medicine, and how this fits into a larger program of topological and geometric approaches to the life sciences.