Analytic geometry (over archimedean and non-archimedean fields)

Berkovich's theory of analytic spaces generalizes the notion of complex manifolds, where the field of complex numbers is replaced by a general valued field, (for example, p-adic numbers or Laurent series). The general theory has applications in various fields including arithmetic/Arakelov geometry, tropical geometry and combinatorics, complex geometry, dynamics, etc. This talk will give a gentle introduction to the subject. If time permits, I might even talk about some applications or connections to other areas of mathematics.