Hausel, TamásWyss, Dimitri Stelio2017-10-272017-10-272017-10-27201710.5075/epfl-thesis-8025https://infoscience.epfl.ch/handle/20.500.14299/141655urn:nbn:ch:bel-epfl-thesis8025-1In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the $p$-adic pushforward of the Haar measure under a hypertoric moment map $\mu$. This leads to an explicit formula for the Igusa zeta function $\FI_\mu(s)$ of $\mu$, and in particular to a small set of candidate poles for $\FI_\mu(s)$. We also study various properties of the residue at the largest pole of $\FI_\mu(s)$. Finally, if $\mu$ is constructed out of a quiver $\Gamma$ we give a conjectural description of this residue in terms of indecomposable representations of $\Gamma$ over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or $p$-adic volume computation, which is only possible due to some surprising cancellations. These cancellations are reminiscent of a result in classical symplectic geometry by Duistermaat and Heckman on the localization of the Liouville measure, hence the title of the thesis.enHypertoric VarietiesNakajima Quiver VarietiesOpen de Rham SpacesIgusa Zeta FunctionsMotivic Fourier TransformDuistermaat-Heckman Theoremthesis::doctoral thesis