Genericity and classes of ultradifferentiable functions

As surprising as it may seem, there exist infinitely differentiable functions which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A classical technique is to use the Baire category theorem and the notion of residuality. This notion is purely topological and does not give any information about the measure of the set of objects satisfying such a property. In this purpose, the notion of prevalence has been introduced. Moreover, one could also wonder whether large algebraic structures of such objects can be constructed. This question is formalized by the notion of lineability.
The first objective of this talk is to go further into the study of nowhere analytic functions. It is known that the set of nowhere analytic functions is residual and lineable in C^infty([0, 1]). We prove that the set of nowhere analytic functions is also prevalent in this space. Those results of genericity are then generalized using Gevrey classes, which can be seen as intermediate between the space of analytic functions and the space of infinitely differentiable functions. We also study how far such results of genericity could be extended to spaces of ultradifferentiable functions, defined using weight sequences.