Basic Notions: Noncommutative geometry and algebraic geometry over the "field with one element"
Matilde Marcolli (California Institute of Technology)
MSRI: Simons Auditorium
I will review two of the various possible approaches to defining algebraic varieties over the non-existing field F1, one due to Soule' based on cyclotomic points and one due to Borger based on Lambda-rings. I will show, based on joint work with Connes and Consani arXiv:0806.2401, that the noncommutative space of the Bost-Connes quantum statistical mechanical system gives the (pro)-variety over F1 that accounts for the unramified extensions with their Galois action in the sense of Kapranov-Smirnov. I will also show, based on my more recent arXiv:0901.3167, that a simple generalization of the Bost-Connes system gives universal Lambda-rings consistently with Borger's approach to F1 geometry. I will also discuss Manin's proposal of analytic geometry over F1 based on the Habiro ring of analytic functions of roots of unity and some intriguing connections to 3-manifold topology.