Vincent BAGAYOKO
Research
keywords: transseries, surreal numbers, model theory. My PhD subject, supervised by Françoise Point and Joris van der Hoeven, is entitled "Hyperseries and surreal Numbers". My goal is to define and study fields of generalized formal series called "hyperseries". Those comprise transseries, as well as formal objects called hyperlogarithms and hyperexponentials which behave like transfinite iterators of the logarithm and the exponential. For instance, a hyperexponential E satisfies Abel's equation E(x+1)=exp(E(x)). Those fields will enjoy, like the field of transseries, notions of transfinite sums, of derivation and composition of series. One such particular field Hy will be closed under various types of functionnal equations involving the field operations, the derivation and compositions of series. In the meantime, I want to define hyperexponential and hyperlogarithmic functions on the field No of surreal numbers, in order to use No as a model to construct Hy. We expect that Hy and No be naturally isomorphic. This will on the one hand yield a formal model of the asymptotic behavior of sufficiently regular real-valued functions (in particular those which lie in Hardy fields) and to construe surreal numbers as avatars of those functions.