Geometric Motivic Integration on Artin n-stacks

Since its conception by Kontsevich in 1995, thetechnique of motivic integration has found numerousapplications in algebraic geometry and representationtheory. The work of Denef, Loeser and Cluckers led tothe formulation of different versions of motivicintegration - geometric motivic integration,arithmetic motivic integration and the theory of"constructible motivic functions". This bookaddresses the problem of generalizing the theorygeometric motivic integration to Artin n-stacks. Wefollow the construction of higher Artin stacks asproposed by Toen and Vezzosi. A brief review of thisconstruction along with some of the basic notions ofhomotopical algebra is also provided. Applications ofthe theory of motivic integration on varieties havebeen very fruitful and this work should pave the wayfor similar results for Artin stacks. Also, some ofthese ideas may prove useful in generalizing otherversions of motivic integration to Artin stacks.