Obtaining bi-Lipschitz stratifications using valued fields

(Work in progress, together with Yimu Yin)One tool to describe singularities of (e.g. algebraic or analytic) subsets X of R^n or C^n are stratifications: a partition of X into finitely many ?Roestrata?R such that any two points x, y in X within the same stratum have the ?Roesame type of neighbourhood?R. The most classical stratifications are Whitney stratifications, which classify neighbourhoods up to homeomorphism. The strongest known stratifications are Mostowski’s bi-Lipschitz stratifications, which classify neighbourhoods up to a bi-Lipschitz map. I will present a new way of obtaining such bi-Lipschitz stratifications, by working in a suitable valued field (a non-standard model of R or C). Up to recently, the existence of bi-Lipschitz stratifications was only known for sub-analytic sets. Our method works for definable sets in arbitrary polyinomially bounded o-minimal expansions of R, i.e. in particular for quasi-analytic sets. (A similar result has also been announced by Valette.)