On numbers, germs, and transseries

Germs of real-valued functions, surreal numbers, and transseries are three ways to enrich the real continuum by infinitesimal and infinite quantities. Each of these comes with naturally interacting notions of ordering and derivative. The category of $H$-fields provides a common framework for the relevant algebraic structures. We give an exposition of our results on the model theory of $H$-fields, and we report on recent progress in unifying germs, surreal numbers, and transseries from the point of view of asymptotic differential algebra.