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.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/5312
[4] http://www.mpim-bonn.mpg.de/node/9806/program?page=last
[5] http://www.mpim-bonn.mpg.de/node/9806/abstracts