My lecture this year focused on recent results (obtained in collaboration with C. Consani [3], [4], [5], [6]) on the limiting case of the " characteristic 1 ". The main aim is to show that the space of adel classes of a global body, which until now has only been considered as a (non-commutative ) space, in fact admits a natural algebraic structure. We shall also see that the Witt ring construction of a ring of characteristic p > 1 admits an analogue in characteristic 1 and that the deformation of the additive structure crucially involves entropy.
References
[3] Connes A., Consani C., "On the notion of geometry overF1 ", Journal of Algebraic Geometry, arXiv08092926v2 [mathAG].
[4] Connes A., Consani C., "Schemes overF1 and zeta functions", Compositio Mathematica, arXiv:0903.2024v3 [mathAG,NT].
[5] Connes A., Consani C., "Characteristic 1, entropy and the absolute point, arXiv:0911.3537v1 [mathAG].
[6] Connes A., Consani C., "The hyperring of adèle classes", arXiv:1001.4260v2 [mathAG]