## All Dissertations, Theses, and Capstone Projects

#### Title

Motivic integration over nilpotent structures

10-2014

Dissertation

Ph.D.

Mathematics

Hans Schoutens

Mathematics

#### Keywords

Algebraic Geometry, arc spaces, auto Igusa zeta function, deformation theory, Motivic Integration, p-adic analysis

#### Abstract

This thesis concerns developing the notion of Motivic Integration in such a way that it captures infinitesimal information yet reduces to the classical notion of motivic integration for reduced schemes. Moreover, I extend the notion of Motivic Integration from a discrete valuation ring to any complete Noetherian ring with residue field $\kappa$, where $\kappa$ is any field. Schoutens' functorial approach (as opposed to the traditional model theoretic approach) allows for some very general notions of motivic integration. However, the central focus is on using this general framework to study generically smooth schemes, then non-reduced schemes, and then, finally, formal schemes. Finally, a computational approach via Sage for computing the equations defining affine arc spaces is introduced and implemented.

COinS