A classical problem in geometry is the following: can we stratify non smooth spaces (e.g. algebraic varieties, quotients of smooth manifolds by group actions, ...) into smooth strata, in such a way that good “equisingularity conditions” for strata are matched? The notion of Whitney stratification arises to give an answer to this question, and indeed algebraic varieties, analytic varieties, semialgebraic sets and semianalytic sets admit a Whitney stratification. The notion of Whitney stratification is not intrinsic, in that it depends on an embedding of the space in $\mathbb{R}^N$.
On the other hand, the notion of conically smooth structure was introduced by Ayala, Francis and Tanaka in 2017. We will explain how this latter notion is an “intrinsic” version of specifying a Whitney stratification, i.e. it does not depend on the choice of an embedding into $\mathbb{R}^N$. In particular, we show that a Whitney stratified space always admits a canonical conically smooth structure. If time permits, we will provide an application of this result: namely, the affine Grassmannian associated to a reductive group, which is a fundamental object in the Geometric Langlands Program, is a conically smooth space. We will also illustrate other basic features of the notion of conically smooth structure.
This is joint work with Marco Volpe (University of Regensburg).