**Topological Descriptors of Reduced Energy Landscapes**

Using the reduced energy landscape produced in Thrust 1, we will compute its topological and geometric features. The topology and geometry of an energy landscape encode how stable its local minima, its basins of attraction, and its minimal energy paths will be under perturbations of the chemical system. In particular, these topological and geometric descriptors will be used as input for machine learning talks in which we predict the chemical outcomes based upon how the energy landscape transforms as a result of modifications to the chemical environment.Persistent homology provides a global summary of the number of holes of each dimension in the energy landscape, and furthermore, a measure of the robustness of each such topological feature as the energy barrier increases. Much of the popularity of topological data analysis relies on the fact that persistent homology is computable. Morse theory is the standard tool in mathematics to study the shape of an energy function in terms of its critical points: minima, saddle points, and maxima. An assumption of classical Morse theory is that the energy function has only non-degenerate critical points. If this assumption is not fulfilled, another area from (differential) topology and singularity theory can be applied, namely catastrophe theory. Catastrophe theory is used to study of how points of stability change under perturbations by external parameters. An interactive demonstration of a catastrophe can be found here.

Thrust 2 Team Leads:

**Henry Adams**, Mathematics, Colorado State University

**Markus Pflaum**, Mathematics, University of Colorado Boulder

Image: Persistent Homology Barcodes. The 0-, 1-, 2-, and 3-dimensional sublevelset persistent homology barcodes for the hexane energy landscape are shown in red, green, blue, and purple, respectively.