We will explain a new setting of genuine equivariant homotopy theory for infinite groups (like the integers) inspired by work of Kaledin. This is different from the usual approach and allows to prove a comparison to a new relative version of cyclotomic spectra that we introduce (called polygonic spectra). We shall also explain the relation to TR and to Witt vectors with coefficients. This is joint work with A. Krause and J. McCandless.

(joint with Bhargav Bhatt) We prove that the space of $W(k)$-lattices in $W(k)[1/p]^n$, for a perfect field $k$ of characteristic $p$, has a natural structure as an ind-(perfect scheme). This improves on recent results of Zhu by constructing a natural ample line bundle on the space of such lattices.