Résume | The sets definable in an o-minimal expansion of the real field share many topological regularity properties with the real algebraic sets. For this reason, I will say that a real object (a subset of R^n, a real function, or a family thereof) is "tame" if it is definable in some o-minimal structure. Given two tame objects, it is not always possible to make them coexist in some common o-minimal expansion of the real field (in which case we say that the two tame objects are "incompatible").
I will recall the basic properties of tame objects and some of the known (in)compatibility results. The restrictions to the positive real axis of the Riemann Zeta function and of Euler's Gamma function have been known to be tame for some time, but the question of their compatibility remained open. I will talk about recent work of Rolin, Speissegger and myself, where we construct an o-minimal expansion of the real field in which these two functions are definable. In order to achieve this goal, we develop a theory of Borel-Laplace multi-summability for certain power series with real exponents. We use this to produce suitable collections of quasianalytic algebras of real germs in several variables, which are closed under the operations (composition, division, blow-ups, implicit functions) that ensure the o-minimality of the structure they generate (by general results of Rolin and myself, 2015, and of van den Dries and Speissegger, 2000). |