Files in this item
Files  Description  Format 

application/pdf 3392025.pdf (2MB)  (no description provided) 
Description
Title:  Nonstandard Methods in Lie Theory 
Author(s):  Goldbring, Isaac Martin 
Doctoral Committee Chair(s):  van den Dries, Lou 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  In this thesis, we apply model theory to Lie theory and geometric group theory. These applications of model theory come via nonstandard analysis. In Lie theory, we use nonstandard methods to prove two results. First, we give a positive solution to the local form of Hilbert's Fifth Problem, which asks whether every locally euclidean local topological group is locally isomorphic to a Lie group. In connection with the local form of Hilbert's Fifth Problem, we study local groups with a local automorphism whose iterates pointwise approach the trivial endomorphism. Secondly, we prove a generalization of a theorem of Pestov regarding BanachLie algebras. Call a BanachLie algebra enlargeable if it is the Lie algebra of a BanachLie group. Pestov used nonstandard methods to prove that a BanachLie algebra is enlargeable if it possesses a directed family of "uniformly enlargeable" subalgebras whose union is dense. We prove an analogue of this result for a wider class of infinitedimensional Lie algebras, namely the locally exponential Lie algebras. In geometric group theory, we give a nonstandard treatment of the theory of ends developed by Hopf and Freudenthal. 
Issue Date:  2009 
Type:  Text 
Language:  English 
Description:  119 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2009. 
URI:  http://hdl.handle.net/2142/86930 
Other Identifier(s):  (MiAaPQ)AAI3392025 
Date Available in IDEALS:  20150928 
Date Deposited:  2009 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois