Title: Synthetic approaches to equivariant homotopy theory
Abstract: Synthetic homotopy theory is a general framework for constructing interesting contexts for doing homotopy theory: using the data of a spectral sequence in some category $\mathcal{C}$, one can construct another category which can be viewed as a deformation of $\mathcal{C}$. The motivating example of such a theory (due to Gheorghe, Wang, and Xu) is ($p$-complete, cellular) $\mathbb{C}$-motivic spectra, which is a deformation of $\mathcal{C}=\mathrm{Sp}$. Burklund, Hahn, and Senger showed that $\mathbb{R}$-motivic homotopy theory is a deformation of the category of $C_2$-equivariant spectra. I will discuss work in progress to construct deformations of $G$-equivariant homotopy theory for other groups $G$. This is joint with Gabriel Angelini-Knoll, Mark Behrens, Hana Jia Kong, and Maxwell Johnson.