Homotopy types and geometries below Spec$(\mathbb{Z})$ (Livestream from PI)

This talk is based on joint work with Yuri Manin. The idea of a "geometry over the field with one element $\mathbb{F}_1$'' arises in connection with the study of properties of zeta functions of varieties defined over $\mathbb{Z}$. Several different versions of $\mathbb{F}_1$ geometry (geometry below Spec$(\mathbb{Z})$ have been proposed over the years (by Tits, Manin, Deninger, Kapranov--Smirnov, etc.) including the use of homotopy theoretic methods and ``brave new algebra'' of ring spectra (Toën--Vaquié). We present a version of $\mathbb{F}_1$ geometry that connects the homotopy theoretic viewpoint, using Zakharevich's approach to the construction of spectra via assembler categories, and a point of view based on the Bost--Connes quantum statistical mechanical system, and we discuss its relevance in the context of counting problems, zeta-functions and generalised scissors congruences.

(Livestream from Perimeter Institute)