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.
","abstract_html":"<p>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. </p>","abstract_has_math":true,"creators":["Stout, Andrew Ryan"],"institution":"The Graduate School and University Center of The City University of New York","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Hans Schoutens"],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-10-01T07:00:00Z","date_published":"2014-10-01T07:00:00Z","updated_at":"2026-07-24T01:58:39Z","subjects":["Mathematics","Algebraic Geometry","arc spaces","auto Igusa zeta function","deformation theory","Motivic Integration","p-adic analysis"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://academicworks.cuny.edu/gc_etds/499","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Hans Schoutens"]},{"key":"dc:creator","label":"Author","values":["Stout, Andrew Ryan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2001-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["The Graduate School and University Center of The City University of New York"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Algebraic Geometry","arc spaces","auto Igusa zeta function","deformation theory","Motivic Integration","p-adic analysis"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://academicworks.cuny.edu/gc_etds/499"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Motivic integration over nilpotent structures"]}]}],"canonical_facts":{"dc:contributor.advisor":["Hans Schoutens"],"dc:creator":["Stout, Andrew Ryan"],"dc:date.available":["2001-01-01T08:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://academicworks.cuny.edu/gc_etds/499"],"dc:subject":["Mathematics","Algebraic Geometry","arc spaces","auto Igusa zeta function","deformation theory","Motivic Integration","p-adic analysis"],"dc:title":["Motivic integration over nilpotent structures"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["The Graduate School and University Center of The City University of New York"]},"updated_at":"2026-07-24T01:58:39Z"}