Speaker: Miguel Barrero (Radboud University Nijmegen, The Netherlands)

Abstract:
In homotopy theory, $E_\infty$  encode operations that are associative, commutative and unital up to all higher homotopies, and all $E_\infty$-operads are equivalent. In equivariant homotopy theory, when considering group actions, the situation becomes more interesting. The equivariant $\Nin$-operads of Blumberg and Hill are $E_\infty$-operads after forgetting the group actions, but they are not necessarily equivalent as equivariant operads. They represent intermediate notions of commutativity, characterized by the existence of certain transfer
maps.

In this talk I will introduce global $N_\infty$-operads, the globally equivariant analogs of $\Nin$-operads. Global equivariant homotopy theory is the study of global spaces and global spectra, which have compatible actions by all compact Lie groups at the same time. I will discuss how global $N_\infty$-operads are related to $N_\infty$-operads for a single group, and how to classify global $N_\infty$-operads in terms of the transfer maps that they encode. Throughout the talk we will use various versions of the little disks operad as examples of equivariant $N_\infty$-operads and global $N_\infty$-operads.