Topological methods in the study of energy landscapes

Differential topological methods such as Morse and catastrophe theory have been used for many years in mathematics to study the topology of manifolds and to describe and classify singularities. Over time it turned out that both methods can be applied in the sciences as well. For example, Morse theory is now a fundamental tool in topological quantum field theory, and singularity theory (to which catastrophe theory belongs) has been used to describe band crossings in topological insulators. In the talk, it will be explained how Morse and singularity theory can be applied to examine energy landscapes as they appear in physical chemistry. The energy function describing the energy landscape corresponds exactly to what mathematicians call a Morse function. Changing the cutoff energy changes at certain points, the so-called critical points, the topology of the landscape. Introducing appropriate descriptors of the various topologies which appear and finding the critical points via topological or geometric data analysis is the overall goal of the project.