In recent years, Clausen and Scholze introduced condensed mathematics, a new framework for analytic geometry that unifies Archimedean and non-Archimedean geometry and allows for Abelian categories of complete modules. In this talk, I will give a gentle introduction to this theory and explain how it can be used to rediscover the classical notion of a tempered holomorphic function.

A holomorphic function is tempered if it does not grow too fast near the boundary of its domain. For example, the function 1/z is tempered on the punctured unit disc, while exp(1/z) is not. Such functions play a key role in the theory of differential equations, particularly in the quest to find Riemann-Hilbert correspondences, a story that goes back to Hilbert's 21st Problem. Even though this notion is very concrete and genuinely analytic, we will see that condensed mathematics enables us to recover it in a categorical and largely "analysis-free" way.