Some Homotopical Aspects of the Field with One Element

First, I will give a brief review of Connes and Consani's model for algebra and geometry over the field with one element, with a focus on its similarity to Segal's model for connective spectra and algebraic K-theory. I will state some results from joint work with S. Nakamura (UC-Irvine) which suggest, along with the existing results of Connes and Consani, that this model is indeed the "right" one. Lastly, I will describe preliminary work, joint with J. Moeller (Caltech), M. Rovelli (UMass-Amherst) and V. Ozornova (Bonn), on constructing a theory of "stable homotopy theory in characteristic one." In particular, I will point out that the infinite delooping of 𝔽₁ itself (i.e. its Eilenberg-MacLane spectrum) localizes to the classical sphere spectrum.