2026-09-22T10:27:37Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/116162 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Fernandes, Rui Loja Kerman, Ely Lerman, Eugene Pascaleff, James 2022-08 application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms The student, Yuxuan Zhang, accepted the attached license on 2022-06-15 at 10:09. The student, Yuxuan Zhang, submitted this Dissertation for approval on 2022-06-15 at 10:17. This Dissertation was approved for publication on 2022-06-27 at 09:49. DSpace SAF Submission Ingestion Package generated from Vireo submission #18077 on 2022-11-15 at 17:37:39 Local symplectic groupoids and the SGA equation Zhang, Yuxuan 2022-06-27 local symplectic groupoids integrability of Poisson manifolds SGA equation This thesis discusses several problems related to local symplectic groupoids. In Chapter 1, we prove that if a local symplectic groupoid has uniformly discrete associators, then its associative completion is a symplectic groupoid. It follows that a Poisson manifold is integrable if and only if any of its local integrations has uniformly discrete associators. In Chapter 2, we construct a local symplectic groupoid integrating the Heisenberg-Poisson manifold which is not 6-associative. In Chapter 3, we give the conditions for a function to be the generating function for some local symplectic groupoid structure on the cotangent bundle, both for a coordinate space and for an abstract manifold. We also compare different notions of generating functions and analyze the role of the SGA equation. In Chapter 4, we show that the algebraic equation in the SGA equation is equivalent to a groupoid 2-cocycle condition. Under mild assumptions on the local symplectic groupoid, we find a groupoid 2-cocycle which under the van Est map yields the underlying Poisson bivector. Thesis eng https://hdl.handle.net/2142/116162 Copyright 2022 Yuxuan Zhang Ph.D. Dissertation Mathematics University of Illinois at Urbana-Champaign Mathematics