turkmath.org

A $C_2$-graded group is a pair: a group $G$ and its index two subgroup $H$.
Its Real representation is a complex representation of $H$ with an action of the other coset $G\backslash H$ of odd elements in another way that needs to be chosen. Different choices lead to different theories.
Such representations appeared independently in three different disciplines: Algebra, Physics and Topology.
The goal of the talk is to review the formalism and various choices, including resulting theories.
The talk is based on my recent works with James Taylor (Oxford) and Matthew B. Young (Utah State).