This work consists of two independent chapters:

\r\n\r\nThe first is a study of commuting countable Borel equivalence relations, where two equivalence relations R and 5 are said to commute if, as binary relations, they commute with respect to the composition operator , i.e., R \u25e6 S = S \u25e6 R. The primary problem considered is, to what extent does the complexity of E = R V S depend on the complexity of R and S , if R and S commute? This is considered both in the case where the underlying space supports no E-invariant probability measure, and the case where it supports at least one such measure. In the first case, the answer is 'not very much': any such aperiodic equivalence relation E can be written as R V S, where Rand 5 are smooth aperiodic. In the second case, we frame our study within the context of costs, a system of invariants for countable Borel equivalence relations with invariant probability measures, developed by G. Levitt [12] and D. Gaboriau [5]. One aspect of costs which is not well understood is the extent to which 'commutativity' within an equivalence relation (in a more general sense than the definition given above) trivializes its cost. We have shown that, under certain conditions, this is in fact the case. One of the consequences of these investigations is a new, elementary proof of the fact the group SL_2 (Z[^1_2]) is anti-treeable.

\r\n\r\nThe second chapter is motivated by the well known theorem of descriptive set theory that every \u041f^1_1 subset of a Polish (separable, completely metrizable) space admits a \u041f^1_1 scale. We construct a \u041f^1_1 scale on the set of differentiable functions with domain [0,1], which is a \u041f^1_1 subset of the Polish space C([0,1]) . This construction is based on the \u041f^1_1 rank of differentiable functions given by Kechris and Woodin in [4], and, like this rank, is meant to reflect the intrinsic nature of DIFF, and so give a 'natural ' criterion for determining whether the uniform limit of differentiable functions is itself differentiable. We then attempt to further analyze this 'scale criterion' for a sequence of differentiable functions (\u0192_n) by comparing it to the criterion that the sequence (\u0192'_n) converges.

\r\n", "date": "2002", "date_type": "degree", "id_number": "CaltechTHESIS:01262012-103823681", "refereed": "FALSE", "official_url": "https://resolver.caltech.edu/CaltechTHESIS:01262012-103823681", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "collection": "CaltechTHESIS", "reviewer": "Tony Diaz", "deposited_on": "2012-01-26 19:52:01", "doi": "10.7907/RHBN-ZN06", "divisions": { "items": [ "div_pma" ] }, "institution": "California Institute of Technology", "thesis_type": "phd", "thesis_advisor": { "items": [ { "id": "Lorden-G-A", "name": { "family": "Lorden", "given": "Gary A." }, "role": "advisor" }, { "id": "Kechris-A-S", "name": { "family": "Kechris", "given": "Alexander S." }, "role": "co-advisor" } ] }, "thesis_committee": { "items": [ { "id": "Kechris-A-S", "name": { "family": "Kechris", "given": "Alexander S." }, "role": "chair" }, { "id": "Clemens-J-D", "name": { "family": "Clemens", "given": "John D." }, "role": "member" }, { "id": "Luxemburg-W-A-J", "name": { "family": "Luxemburg", "given": "W. A. J." }, "role": "member" }, { "id": "Ramakrishnan-D", "name": { "family": "Ramakrishnan", "given": "Dinakar" }, "orcid": "0000-0002-0159-087X", "role": "member" }, { "id": "Lorden-G-A", "name": { "family": "Lorden", "given": "Gary A." }, "role": "member" } ] }, "thesis_degree": "PHD", "thesis_degree_grantor": "California Institute of Technology", "thesis_defense_date": "2001-12-05", "review_status": "approved", "option_major": { "items": [ "math" ] }, "copyright_statement": "Author's Rights Authorization: I hereby certify that, if appropriate, I have obtained a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted here is the same as that approved by my advisory committee.\n\nI hereby grant to California Institute of Technology or its agents the non-exclusive license to archive and make accessible, under the conditions specified under \"Thesis Availability\" in this submission, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation, or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report.", "resource_type": "thesis", "pub_year": "2002", "author_list": "Pavelich, Janet Mary" } ]