The moduli space of marked singularities

The moduli space of marked singularities was introduced by Claus Hertling in 2010 and parameterizes μ-homotopic isolated hypersurface singularities equipped with certain markings. This moduli space can be understood either as a global μ-constant stratum or as a Teichmüller space of singularities. The additional marking allows one to formulate the conjecture on the analytic behavior of singularities within a distinguished μ-homotopy class in terms of a Torelli type problem in a very efficient way. In my talk I will discuss the history of this problem and introduce carefully the notion of a marked singularity. Secondly, I will speak about very recent results on this Torelli type problem for singularities of high modality.