Classification problems are ubiquitous in mathematics; and in study of Riemannian manifolds with positive sectional curvature, classification is a central theme. In 2003, Wilking introduced a set of topological tools to the study of positive sectional curvature which led to new classification results in the presence of torus symmetry.

This talk will overview some classical results in positive sectional curvature, before introducing Wilking's tools and discussing their applications. We will then see recent results in positive sectional curvature due to Kennard, Khalili Samani, and Searle in the case of discrete symmetries, and I will finish by presenting some of my own results.