A long and winding road to definable sets

We survey the development of o-minimal structures from a geometric point of view and compare them with subanalytic sets insisting on the differences. The idea is to show the long way from semi-analytic to definable sets, from normal partitions to cell decompositions. Some recent results are discussed in the last section

Medial axis and singularities

We correct one erroneous statement made in our recent paper "Medial axis and singularities".Comment: Some minor misprints are corrected and one final remark is adde

Deformation of geometry and bifurcation of vortex rings

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.Comment: 26 page

Stratifications, Equisingularity and Triangulation

International audienceThis text is based on 3 lectures given in Cuernavaca in June 2018 about stratifications of real and complex analytic varieties and subanalytic and definable sets. The first lecture contained an introduction to Whitney stratifications, Kuo-Verdier stratifications and Mostowski's Lipschitz stratifications. The second lecture concerned equisingularity along strata of a regular stratification for the different regularity conditions: Whitney, Kuo-Verdier, and Lipschitz, including thus the Thom-Mather first isotopy theorem and its variants. (Equisingularity means continuity along each stratum of the local geometry at the points of the closures of the adjacent strata.) A short discussion follows of equisingularity for complex analytic sets including Zariski's problem about topological invariance of the multiplicity of complex hypersurfaces and its bilipschitz counterparts. In the real subanalytic (or definable) case we mention that equimultiplicity along a stratum translates as continuity of the density at points on the stratum, and quote the relevant results of Comte and Valette generalising Hironaka's 1969 theorem that complex analytic Whitney stratifications are equimultiple along strata. The third lecture provided further evidence of the tameness of Whitney stratified sets and of Thom maps, by describing triangulation theorems in the different categories, and including definable and Lipschitz versions. While on the subject of Thom maps we indicate examples of their use in complex equisingularity theory and in the definition of Bekka's (c)-regularity. Some very new results are described as well as old ones