Title: Projective geometry as absolute geometry
Abstract: In the 1980’s, Yuri Manin suggested that Deligne’s approach to the function field Riemann Hypothesis might be adopted to the rational numbers by thinking of Spec(ℤ) as a curve over a phantasmic “Spec(𝔽₁).” He referred to algebraic geometry over a “field with one element” as absolute geometry. Since then, many models for a “category of vector spaces over 𝔽₁” have been suggested. This talk will begin by reviewing Alain Connes and Caterina Consani’s highly category theoretic approach to absolute geometry (and will therefore contain a crash course in category theory). This approach has allowed them to prove a version of the Grothendieck-Riemann-Roch Theorem over Spec(ℤ).
Of course, the idea of a “field with one element” did not originate with Manin. It was first suggested in the 1950’s by Jacques Tits in an entirely different context. Tits noticed that certain formulae in projective geometry over the field with q elements continued to make sense if one replaced q with 1 throughout. Moreover, they recovered well-known formulae from combinatorics. He proposed that combinatorics be thought of as a kind of “projective geometry over a field with one element.” In particular, GLₙ(𝔽₁) should be the symmetric group on n elements. The second part of my talk will describe recent joint work with So Nakamura that embeds projective geometry, in the classical axiomatic sense of Veblen, into Connes and Consani’s model for algebra over 𝔽₁.