Allen Gehret : The asymptotic couple of the field of logarithmic transseries

Posted by Kostyantyn Slutskyy , part of the Logic Seminar.

At

Feb. 3, 2015, 4 p.m.

In

SEO 427

Abstract

The differential-valued field $\mathbb{T}_{\log}$ of logarithmic transseries is conjectured to have good model theoretic properties. As a partial result in this direction, and as a confidence building measure we prove that at least its \emph{asymptotic couple} has a good model theory. The value group $\Gamma_{\log}$ of $\mathbb{T}_{\log}$ can be given the additional structure of a map $\psi:\Gamma\to\Gamma$ which is induced by the derivation on $\mathbb{T}_{\log}$. The structure $(\Gamma_{\log},\psi)$ is the asymptotic couple of the field of logarithmic transseries (in the sense of Rosenlicht). In this talk we will discuss the good model-theoretic properties of $(\Gamma_{\log},\psi)$, including a quantifier-elimination result in an appropriate first-order language, definable functions on a certain discrete set, a stable embedding result, and NIP (the Non-Independence Property). All results in this talk (besides NIP) are in <a href="http://arxiv.org/abs/1405.1012">http://arxiv.org/abs/1405.1012</a>.