A real analytic function can always be continued holomorphically to some domain. However, the holomorphic continuations of definable functions in an o-minimal structure may not be definable. I will present joint work with P. Speissegger in which we study holomorphic continuations of functions definable in two o-minimal expansions of the real field. I will also discuss how to apply these results to the complex Gamma function and Riemann zeta function.[-]

As exemplified by o-minimality, imposing strong restrictions on the complexity of definable subsets of the affine line can lead to a rich tame geometry in all dimensions. There has been multiple attempts to replicate that phenomenon in non-archimedean geometry (C, P, V, b minimality) but they tend to either only apply to specific valued fields or require geometric input. In this talk I will present another such notion, h-minimality, which covers all known well behaved characteristic zero valued fields and has strong analytic and geometric consequences. By analogy with o-minimality, this notion requires that definable sets of the affine line are controlled by a finite number of points. Contrary to o-minimality though, one has to take special care of how this finite set is defined, leading to a whole family of notions of h-minimality. This notion has been developed in the past years by a number of authors and I will try to paint a general picture of their work and, in particular, how it compares to the archimedean picture.[-]

In the o-minimal setting, parametrizations of definable sets form a key component of the Pila-Wilkie counting theorem. A similar strategy based on parametrizations was developed by Cluckers-Comte-Loeser and Cluckers-Forey-Loeser to prove an analogue of the Pila-Wilkie theorem for subanalytic sets in p-adic fields. In joint work with R. Cluckers and I. Halupczok, we prove the existence of parametriza- tions for arbitrary definable sets in Hensel minimal fields, leading to a counting theorem in this general context. [-]