| Résume | Introduced by Conway to evaluate partial combinatorial games, surreal numbers consist in a proper class containing "all numbers great and small". Moreover, they can be endowed with a very rich algebraic and even analytic structure, turning them into universal domains for several important theories : linearly ordered sets, ordered Abelian groups, ordered valued fields - in particular ordered generalized series fields via the omega-exponentiation -, real analytic fields and exponential fields, and more recently H-fields (an abstract version of Hardy fields due to M. Aschenbrenner and L. van den Dries).
In this talk, I will introduce these fascinating objects, starting with the very basic definitions, and will give a quick overview, with a particular emphasis on exp (which extends exp on the real numbers) and the omega map (which extends the omega-exponentiation for ordinals). This will help me to subsequently present our recent contributions with A. Berarducci, S. Kuhlmann and V. Mantova concerning the notion of omega-fields (possibly with exp). One of our motivations is to clarify the link between composition and derivation for surreal numbers. |
