Group actions, algebra up to homotopy, and flavors of commutativity

A group is just a set together with a multiplication with certain properties, and these are ubiquitous in modern mathematics. In homotopy theory, we often see less-rigid objects, with many of the properties like associativity or commutativity only holding "up to homotopy". This part of the story has been well-understood since the 1970s.

When we add in the action of a fixed finite group, the game changes wildly. Even what we mean by commutative becomes less clear. This is the heart of the evolving subject of equivariant algebra. I'll describe how to understand groups "up to homotopy", what we see when we put in a group action, and how a classification problem of what "commutative” means in this context connects to combinatorics.