An equisingular heritage of Bernard Teissier
Georges Comte
TL;DR
This article surveys Bernard Teissier's equisingularity program and its expansion to real and non-Archimedean contexts through tame geometry. It emphasizes canonical stratifications (Whitney, and its real/non-Archimedean extensions via t-stratifications) and the role of polar invariants and local multiplicities in certifying regularity along strata. Real and non-Archimedean theories replace complex multiplicities with density and polar-invariant sequences such as $\Theta_d$ and $\sigma_i$, and introduce tools like the Riso-tree to structure equisingularity in valued fields. The work highlights the cross-pollination between algebraic ideas and metric/differential approaches, extending Teissier's legacy to broad tameness frameworks and motivating future connections to curvature and motivic invariants.
Abstract
We give a brief and partial overview of Bernard Teissier's work in complex equisingularity theory, and a perspective on its legacy; in particular, we focus on the development of the theory in the real and the non-Archimedean contexts. Our aim is not to go into technical details, but, hopefully, rather to give a flavour of the forms taken by these developments, while providing enough definitions and references to give the reader access to old and new reference articles in the field.