Lipschitz geometry of singularities – LISA

This project focuses on the very active field of Lipschitz geometry of singularities. Its essence is the following natural problem. It has been known since the work of Whitney that a real or complex algebraic variety is topologically locally conical. On the other hand it is in general not metrically conical: there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. A natural problem is then to build classifications of the germs up to local bi-Lipschitz homeomorphism, and what we call Lipschitz geometry of a singular space germ is its equivalence class in this category. There are different approaches for this problem depending on the metric one considers on the germ. A real analytic space germ (V,p) has actually two natural metrics induced from any embedding in RN with a standard euclidean metric: the outer metric is defined by the restriction of the euclidean distance, while the inner metric is defined by the infimum of lengths of paths in V.
Lipschitz geometry of singular sets is an intensively developing subject which started in 1969 with the work of Pham and Teissier on the Lipschitz classification of germs of plane complex algebraic curves. The project presented here is motivated by several important results obtained in this area during the last decade by Birbrair, Fernandes, Gabrielov, Gaffney, Grandjean,Houston, Lê, Neumann, Parusinski, Paunescu, Pichon, Ruas, Sampaio and others. We think in particular about the surprising discovery by Birbrair and Fernandes that complex singularities of dimension at least two are in general not metrically conical for the inner metric, which started a series of works leading to the complete classification of the inner Lipschitz geometry of germs of normal complex surfaces by Birbrair, Neumann and Pichon (coordinator of the project), and building on it, to major progress in the study of the outer metric. Another important result is the proof by Parusinski (member of our team) and Paunescu of the Whitney fibering conjecture in the analytic setting, based on a relation between Zariski equisingularity and the arc-wise analytic equisingularity which is of similar nature as Lipschitz equisingularity.
Our project has two main objectives: (1) Building classifications of Lipschitz geometry in larger settings such as non-isolated and higher dimensional complex singularities, function germs, and in the global, semi-algebraic and o-minimal settings, (2) Developing bridges between Lipschitz geometry and three other aspects of singularity theory:
- equisingularity;
- embedded topology;
- arc spaces and resolution theory.
These three topics are classical areas of singularity theory, but their relations with Lipschitz geometry remain almost unexplored. However, some results have been obtained in all three areas. For example, a long standing question asking if Zariski equisingularity can be interpreted from a Lipschitz point of view got recently (2014) a positive answer for complex surface singularities, but remains open in higher dimensions. Another example is the recent result that the outer Lipschitz geometry of a normal surface singularity determines its multiplicity. This result gives an approach to the famous Zariski multiplicity question from a Lipschitz point of view which, again, needs to be explored in higher dimensions; it also emphasizes the importance of studying the relations between Lipschitz geometry and embedded topology of a hypersurface. As a last example, recent works show the key role played by wedges of arcs in the Lipschitz classifications of real and complex singularities, and of the resolution of singularities in the complex setting.
In view of these recent results, as well as several other ones, we strongly believe that Lipschitz geometry will give a new point of view on each of them and will help to solve several important open problems which are a priori not of metric nature.