Banach algebraic geometry
I will talk about two topics which give support to a unified theory of archimedean and non-archimedean analytic geometry. In both examples I will review a topic in complex analytic geometry (results from the 1970's) and, after reinterpreting it, show that the same thing happens in non-archimedean geometry (giving new results). The first topic is a non-archimedean version of Ishimura's theorem. This theorem states that on a complex manifold, the continuous linear endomorphisms of the structure sheaf agrees with the sheaf of formal differential operators whose symbol is holomorphic on the cotangent bundle. The second topic is about acyclicity. On a complex analytic space, this is about "quasi-coherent sheaves" not having higher cohomology on Stein spaces. I explain a similar result in the non-archimedean context. The tools used involve an interesting mix of homological algebra and functional analysis. I will explain some potential applications of both of these topics related to number theory. No knowledge about cohomology, differential operators, Stein spaces, or any sort of analytic geometry will be assumed.