Block Vorlesung on Transseries
This is a series of ten lectures about transseries.
At a concrete level, transseries are formal generalized power series involving a variable x seen as a generic positive infinite element of a non-Archimedean ordered field, and exponentials, logarithms, and combinations thereof. They were introduced independently by Dahn-Göring (1987) and Ecalle (1992) respectively in order to study Tarski's problem on the decidability of the real exponential field, and Dulac problem on limit cycles of polynomial or analytic vector fields, respectively.
There are strong connections between transseries are strongly and the model theory of ordered exponential fields, that of ordered valued differential fields (like Hardy fields), and to problems in real analytic geometry. In these lectures, I will give an introduction to the main features of a more abstract notion of transseries introduced by Michael Schmeling and Joris van der Hoeven,
- Construct small and very large fields of transseries.
- Draw connections between fields of transseries and models of real exponentiation.
- Study a convex equivalence relation on ordered exponential fields, coarser than Archimedean equivalence, which is crucial in understanding the complex algebraic structure of transseries fields
- Introduce the notion of (infinitely) nested transseries, and see how it relates to the existence of maximal transseries fields with
- Define derivations and composition laws on certain fields of transseries containing nested transseries.
In the course of this, I will introduce the notion of fields of well-based series (an equivalent reformulation of the notion of Hahn series field ), of summability of transfinite families in such fields, and of analyticity of functions on such fields.
Scripts for the lectures
Lecture 1: Well-founded, well-based, and Noetherian classes
Lecture 2: Hahn power groups and fields of well-based series
Lecture 3: Formally analytic functions
Lecture 4: Level structures and log-fields
Lecture 5: Transserial fields
Lecture 6: Transserial calculus
Lecture 7: Nesting over transserial fields
Lecture 8: Nested extensions
Lecture 9: A derivation on maximal transseries fields
Lecture 10: A composition law on a maximal transseries field
Letzte Änderung: