Berkovich spaces

Berkovich spaces are advanced mathematical structures used in non-Archimedean geometry to study solutions of equations over fields with special types of absolute values, like p-adic numbers. Unlike traditional geometric spaces, they incorporate not just points but also limit behaviors and types of "degenerate" solutions. This creates a richer, more flexible space that helps mathematicians understand properties of equations and functions in contexts where usual geometry falls short. In essence, Berkovich spaces provide a framework to analyze arithmetic and geometric questions over fields with non-standard valuations, bridging algebra, analysis, and topology in a unified way.