Liquid modules and complex analysis

I will report on joint work with Peter Scholze in which we define and study the concept of "liquid real vector space".  This is a replacement for the classical notion of complete locally convex real vector space, designed to have excellent formal algebraic properties while still admitting a richness of examples relevant to analytic geometry.  I will explain the basics of the liquid theory, and as an application describe how it yields a theory of "quasicoherent sheaves" on complex analytic spaces, generalizing the classical theory of coherent sheaves to the setting of infinite-dimensional fibers.