At the end of the 19^th century two types of mathematical infinities were introduced: du Bois-Reymond developed a “calculus of infinities” to deal with the “regular” growth rates of real functions at infinity, whereas Cantor proposed his theory of ordinal numbers as a way to count beyond all natural numbers.

Transseries and surreal numbers provide two modern generalizations of these theories. Transseries are formal objects that cover most regular growth rates that can be encountered at infinity. Conway's surreal numbers form the ultimate extension of the field of real numbers with infinitely large and small quantities and in particular with all Cantor's ordinal numbers.

Together with Bagayoko, Schmeling, Kaplan, and van den Dries, I have developed a further (and ultimate) generalization of transseries, called hyperseries, with the aim of covering all regular formal growth rates of functions at infinity.

It turns out that there is a remarkable correspondence between surreal numbers and hyperseries: any surreal number can be written uniquely as the value of a hyperseries at the first infinite ordinal . Current work in progress concerns the definition of a derivation and a composition on hyperseries in an infinitely large indeterminate .Via the above correspondence, this should also lead to a canonical derivation and composition on the field of surreal numbers.

Occasion: Mini-workshop on Transseries and Dynamical Systems, Fields Institute, Toronto, June 1, 2022

Documents: slideshow, TeXmacs source