Complex analytic geometry is among the most successful fusions of analysis and algebra in mathematics, and much of this is reliant on the fact that the usual absolute value on \$\mathbb{C}\$ is highly unique. But what happens when we try to do analytic geometry with other absolute values or other fields? In the (more-typical) non-archimedean situation, the metric spaces defined on rational points are particularly badly-behaved, and we are in need of a "fix" to have a coherent notion of analytic geometry. We survey some of these pathological properties, describe how they can be remedied using inspiration from functional analysis, and see how this remedy affords us powerful analytic techniques familiar from the complex setting.
Zoom Link: https://washington.zoom.us/j/92849568892