M/E23

The class of surreal numbers was invented and first described by Conway [1]. This class has the property that it contains both the set of real numbers and the class of ordinal numbers. The operations of addition and multiplication as well as a total order relation arise naturally on the surreal numbers, making them a (linearly) ordered field (of class size). Another way of describing surreal numbers is by expressing them as generalised power series (i.e., Hahn series) via their Conway normal form. This leads to a valuation theoretic approach towards the study of surreal numbers: the ordered field of surreal numbers can be regarded as a Hahn field whose residue field is given by the real numbers and its value group by the surreal numbers. In my talk, I will first outline Conway's surprisingly simple but powerful construction of surreal numbers from first principles. Moreover, I will explain what Hahn fields are and how surreal numbers can be regarded as generalised power series. If time permits, I will report on joint work in progress with E. Kaplan and M. Serra building on the study of automorphisms of Hahn fields initiated in by Kuhlmann and Serra [2]. [1] J. H. Conway, On numbers and games, Lond. Math. Soc. Monogr. 6 (Academic Press, London, 1976). [2] S. Kuhlmann and M. Serra, The automorphism group of a valued field of generalised power series, J. Algebra 605 (2022) 339–376.

Lothar Sebastian Krapp

Konstanz