Topology is dedicated to the study of shapes, and its starting point is an easy-sounding question: How can I tell if two objects are similar? While humans are very adept at distinguishing a large variety of shapes, it is not always easy to say precisely what makes this object similar to or distinct from that one. For example, comparing a sphere to a torus we may observe that a torus has a hole while the sphere does not. Topology makes such observations precise and provides computational methods that can capture similarity of objects numerically. One way to do that is to use persistent homology, which has been developed over the last decade or so specifically to aid numerical characterisation of complex objects.

The talk will introduce basic ideas and will demonstrate their power on a main example, which is the study of the topology of human lungs in COPD, the chronic obstructive pulmonary disease. COPD is the third leading cause of death worldwide. Despite its prevalence and importance, quantitative diagnostic methods are rather limited and rely on expert input. In particular, computer tomography (CT) scans are widely used, but there is no accepted standard procedure that would lead to numerical outputs that would allow classification and comparison of patients.

In this talk, I will present the results of our recent study in which we develop topological characteristics derived from CT scans that allow for a direct comparisons between patients and provide a basis for classification. We demonstrate that our classification methods are on the one hand strong enough to reproduce known results and on the other are more subtle and can potentially be used to stratify the disease more finely.

To show the flexibility of the topological methods I will also present a contribution to the problem of understanding the topological and geometric structure of chemical compounds that will make them soluble in water. This is of key importance in design new compounds that can be used to create new drugs.

The talk will be accessible to non-specialists.