“Equivariant Homology: Distinguishing Spaces by their Holes, Rotations, and Reflections” 
by Sarah Petersen ’17, PhD

Topology is an area of math which studies sets together with a notion of continuity. The real line is a familiar example where our idea of continuity coincides with having no gaps or places we would lift the pencil when drawing it. Sets together with a notion of continuity are called topological spaces. Other examples of topological spaces include surfaces, spheres, and higher dimensional spaces. A foundational question in mathematics is how objects are either the same or different, and we can ask this question about topological spaces. One geometrically motivated answer and way to distinguish among spaces is to consider the number and types of holes in that space. This motivates the definition of an invariant called homology. Equivariant homology further takes into account rotations and reflections of the space under consideration. This talk will focus on introducing some of the ideas underpinning equivariant homology, starting with the example of a circle reflected across its diameter. Along the way, we will build enough understanding to state a new theorem computing the equivariant homology of certain spaces called equivariant Eilenberg-MacLane spaces.