Files in this item

FilesDescriptionFormat

application/pdf

application/pdfMICHIELS-DISSERTATION-2018.pdf (5MB)
(no description provided)PDF

Description

Title:Symplectic foliations, currents, and local Lie groupoids
Author(s):Michiels, Daan
Director of Research:Loja Fernandes, Rui
Doctoral Committee Chair(s):Tolman, Susan
Doctoral Committee Member(s):Kerman, Ely; Pascaleff, James
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):foliation
symplectic foliation
Poisson structure
current
calibration
structure cycle
local Lie groupoid
associativity
globalizability
associative completion
associators
integrability
monodromy
Abstract:This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham’s theory of currents, and Sullivan’s theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. Next, three main results are obtained. The first is the generalization of a theorem by Mal’cev. The original theorem characterizes the local Lie groups that are part of a (global) Lie group. We give the corresponding result for local Lie groupoids. The second result is the generalization of a theorem by Olver which classifies local Lie groups in terms of Lie groups. Our generalization classifies, in terms of Lie groupoids, those local Lie groupoids that have integrable algebroids. The third and final result demonstrates a relationship between the associativity of a local Lie groupoid, and the integrability of its algebroid. In a certain sense, the monodromy groups of a Lie algebroid manifest themselves combinatorially in a local integration, as a lack of associativity.
Issue Date:2018-04-09
Type:Thesis
URI:http://hdl.handle.net/2142/100940
Rights Information:Copyright 2018 Daan Michiels
Date Available in IDEALS:2018-09-04
Date Deposited:2018-05


This item appears in the following Collection(s)

Item Statistics