Regularity and Nearness Theorems for Families of Local Lie Groups

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dc.contributor.advisorHardt, Robert M.en_US
dc.creatorMcGaffey, Tomen_US
dc.date.accessioned2013-03-08T00:36:34Zen_US
dc.date.available2013-03-08T00:36:34Zen_US
dc.date.issued2011en_US
dc.description.abstractIn this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate development, we construct a nontrivial topology for rings of map germs on Euclidean spaces. First, we develop a framework for the theory of (local) nonstandard Lie groups and within that framework prove a nonstandard result that implies that a family of local Lie groups that converge in a pointwise sense must then differentiability converge, up to coordinate change, to an analytic local Lie group, see corollary 6.3.1. The second result essentially says that a pair of mappings that almost satisfy the properties defining a local Lie group must have a local Lie group nearby, see proposition 7.2.1. Pairing the above two results, we get the principal standard consequence of the above work which can be roughly described as follows. If we have pointwise equicontinuous family of mapping pairs (potential local Euclidean topological group structures), pointwise approximating a (possibly differentiably unbounded) family of differentiable (sufficiently approximate) almost groups, then the original family has, after appropriate coordinate change, a local Lie group as a limit point. (See corollary 7.2.1 for the exact statement.) The third set of results give nonstandard renditions of equicontinuity criteria for families of differentiable functions, see theorem 9.1.1. These results are critical in the proofs of the principal results of this paper as well as the standard interpretations of the main results here. Following this material, we have a long chapter constructing a Hausdorff topology on the ring of real valued map germs on Euclidean space. This topology has good properties with respect to convergence and composition. See the detailed introduction to this chapter for the motivation and description of this topology.en_US
dc.format.extent226 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS MATH. 2011 MCGAFFEYen_US
dc.identifier.citationMcGaffey, Tom. "Regularity and Nearness Theorems for Families of Local Lie Groups." (2011) Diss., Rice University. <a href="https://hdl.handle.net/1911/70349">https://hdl.handle.net/1911/70349</a>.en_US
dc.identifier.digitalMcgaffeyTen_US
dc.identifier.urihttps://hdl.handle.net/1911/70349en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectPure sciencesen_US
dc.subjectLie groupsen_US
dc.subjectRegularity theoremsen_US
dc.subjectHilbert's Fifth problemen_US
dc.subjectTheoretical mathematicsen_US
dc.titleRegularity and Nearness Theorems for Families of Local Lie Groupsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineNatural Sciencesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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