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.[-]

Joint work with Silvain Rideau-Kikuchi.
Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this talk, we propose a unified framework for studying them: the class of pseudo $T$ -closed fields, where $T$ is an enriched theory of fields. These fields verify a 'local-global' principle for the existence of points on varieties with respect to models of $T$ . This approach also enables a good description of some fields equipped with multiple V -topologies, particularly pseudo algebraically closed fields with a finite number of valuations. An important result that will be discussed in this talk is a (model theoretic) classification theorem for bounded pseudo T -closed fields, in particular we show that under specific hypotheses on $T$ , these fields are NTP2 of finite burden.[-]