Symplectic foliations, currents, and local Lie groupoids

Michiels, Daan

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https://hdl.handle.net/2142/100940 Copy

Description

Title

Symplectic foliations, currents, and local Lie groupoids

Author(s)

Michiels, Daan

Issue Date

2018-04-09

Director of Research (if dissertation) or Advisor (if thesis)

Loja Fernandes, Rui

Doctoral Committee Chair(s)

Tolman, Susan

Committee Member(s)

• Kerman, Ely
• Pascaleff, James

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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.

Graduation Semester

2018-05

Type of Resource

text

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http://hdl.handle.net/2142/100940

Copyright and License Information

Copyright 2018 Daan Michiels

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