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Friday, November 18, 2016

**Abstract:** $H$-fields are ordered differential fields which serve as an abstract generalization of both Hardy fields (ordered differential fields of germs of real-valued functions at $+\infty$) and transseries (ordered valued differential fields such as $\mathbb{T}$ and $\mathbb{T}_{\log}$). A *Liouville closure* of an $H$-field $K$ is a minimal real-closed $H$-field extension of $K$ that is closed under integration and exponential integration. In 2002, Lou van den Dries and Matthias Aschenbrenner proved that every $H$-field $K$ has exactly one, or exactly two, Liouville closures, up to isomorphism over $K$. Recently (in arxiv.org/abs/1608.00997), I was able to determine the precise dividing line of this dichotomy. It involves a technical property of $H$-fields called $\lambda$-freeness. In this talk, I will review the 2002 result of van den Dries and Aschenbrenner and discuss my recent contribution.