Laureate 2019

Symposium

Tuesday, 29 October 2019

Simplicity and complexity in model theory

external page Maryanthe Malliaris (University of Chicago)

Abstract: Recent progress in model theory is changing our understanding of how the finite and infinite interact. One aspect of this story has to do with the emerging appearance of complexity within the so-called simple unstable theories, which provide a model theoretic framework for studying certain familes of 'random' objects, such as the theories of random graphs and hypergraphs, as well as pseudofinite fields.

Ample structures of finite Morley rank

external page Katrin Tent (University of Münster)

Abstract: The model theoretic notion of ampleness captures essential properties of projective spaces over fields. From the model theoretic perspective it is natural to ask whether any sufficiently ample structure arises from a projective space and hence from a field. In this talk I will explain the question and present recent results on ample structures.

Hardy fields and transseries

external page Lou van den Dries (University of Illinois)

Abstract: This is joint work with Matthias Aschenbrenner (UCLA) and Joris van der Hoeven (Ecole Polytechnique, Paris). We are aiming for what would be the ultimate extension result for Hardy fields. A $Hardy$ $field$ (Bourbaki) is a differential field of germs at $+\infty$ of differentiable real-valued functions defined on intervals ($a$, $+\infty$), with the derivation given by ordinary differentiation. Let $H$ be a Hardy field. Then $H$ is a totally ordered field by declaring for $f \in H$:

$f > 0 : \Longleftrightarrow f(t) > 0$ eventually, that is, for all sufficiently large $t$.

Let $P(Y)$ be a differential polynomial over $H$, and suppose $P$ changes sign on $H: P(f) < 0 < P(g)$, where $f < g$ in $H$. Then we conjecture that there exists a germ $\phi$ in a Hardy field extension $H$^∗ of $H$ such that $f < \phi < g$ and $P(\phi) = 0$. At the time of writing this abstract the proof of this conjecture is not yet complete. A consequence would be: all maximal Hardy fields are elementarily equivalent to the ordered differential field $\mathbb{T}$ of transseries.