abstract: In this talk, I will talk about the conditional measures of determinantal point processes and its application in resolving the Lyons-Peres completeness conjecture (joint with Alexander Bufetov and Alexander Shamov). The formalism of these conditional measures also allow us to obtain the Palm equivalence of determinantal point processes with Bergman kernels in very general domains of any dimensional complex spaces (joint with Alexander Bufetov and Shilei Fan).