PIRSA:18080052

Homotopy types and geometries below Spec(Z)

APA

Marcolli, M. (2018). Homotopy types and geometries below Spec(Z) . Perimeter Institute for Theoretical Physics. https://pirsa.org/18080052

MLA

Marcolli, Matilde. Homotopy types and geometries below Spec(Z) . Perimeter Institute for Theoretical Physics, Aug. 15, 2018, https://pirsa.org/18080052

BibTex

          @misc{ scivideos_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 for Theoretical Physics},
            year = {2018},
            month = {aug},
            note = {PIRSA:18080052 see, \url{https://scivideos.org/index.php/pirsa/18080052}}
          }
          

Matilde Marcolli University of Toronto

Talk numberPIRSA:18080052
Source RepositoryPIRSA
Talk Type Conference

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.