Homotopy types and geometries below Spec(Z)

          @misc{ pirsa_PIRSA:18080052,
            doi = {10.48660/18080052},
            url = {https://pirsa.org/18080052},
            author = {Marcolli, Matilde},
            keywords = {Mathematical physics},
            language = {en},
            title = { Homotopy types and geometries below Spec(Z) },
            publisher = {Perimeter Institute},
            year = {2018},
            month = {aug},
            note = {PIRSA:18080052 see, \url{https://pirsa.org}}
          }
          

Abstract

This talk is based on joint work with Yuri Manin. The idea of a “geometry over the field with one element F1” arises in connection with the study of properties of zeta functions of varieties defined over Z. Several different versions of F1 geometry (geometry below Spec(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 (To¨en–Vaqui´e). We present a version of F1 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.