The differential field \(\mathbb T \) of transseries is an ordered field extension of the real field \(\mathbb R \) and is a kind of universal domain for asymptotic real differential algebra. It has appeared in many different and rich contexts, e.g. Hilbert’s sixteenth problem. About twenty years of research have culminated in the monumental monograph [M. Aschenbrenner et al., Asymptotic differential algebra and model theory of transseries. Princeton, NJ: Princeton University Press (2017; Zbl 1430.12002)], where the main model-theoretic conjectures have been established. These can be roughly summed up by saying that an enlightening quantifier elimination result is reached.
Some key questions relating the field of transseries to the more classical Hardy fields of germs of real functions were still to be elucidated. This paper is a brief account by L. van den Dries, in his proverbial lucid and enlightening expository style, of the results in this direction in the intervening years, which more or less “complete the picture” drawned in their book. Mostly, they proved results conjectured in their ICM lecture of 2018 [M. Aschenbrenner et al., in: Proceedings of the International Congress of Mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1–9, 2018. Volume II. Invited lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM), 1–23 (2018; Zbl 1445.03044)]. I will only mention the following, mentioned in the paper under review : all maximal Hardy fields are elementarily equivalent to \(\mathbb T\), i.e they have the same first-order theory; in fact, any two maximal Hardy fields are back-and-forth equivalent, “which is considerably stronger than elementarily equivalent: under the Continuum Hypothesis it follows that any two maximal Hardy fields are even isomorphic, as ordered differential fields.”; finally, every maximal Hardy field is back-and-forth equivalent to the field of Conway’s surreal numbers of countable length, equipped with the Berarducci-Mantova derivation.