This thesis comprises four independent parts and an appendix.
\r\n\r\n1. We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure, and category settings, and establish some basic correspondences between the two notions. We then explore in detail many examples, including finding spanning trees in graphs, finding monochromatic sets in Ramsey's Theorem, and linearizing partial orders.
\r\n\r\n2. Standard results in descriptive set theory provide sufficient conditions for a set P \u2286 \u2115\u2115 \u00d7 \u2115\u2115 to admit a Borel uniformization, namely, when P has small or large sections. We consider an invariant analogue of these results with respect to a Borel equivalence relation E. Given E, we show that every such P admits an E-invariant Borel uniformization if and only if E is smooth. We also compute the definable complexity of counterexamples in the case where E is not smooth, using category, measure, and Ramsey-theoretic methods. We also show that the set of pairs (E, P) such that P has large sections and admits an E-invariant Borel uniformization is \u03a312-complete.
\r\n\r\n3. Let E, F be Borel equivalence relations on X, Y, and P be an E-invariant Borel set whose sections contain countably many F-classes. We explore obstructions to the existence of Borel E-invariant uniformizing sets for P, i.e., sets choosing one F-class from every section. We survey known results, and prove new dichotomies for the case where P has \u03c3-bounded finite sections. On the way, we prove a dichotomy characterizing the essential values of Borel cocycles into residually finite Polish groups.
\r\n\r\n4. We show that the Kechris\u2013Solecki\u2013Todor\u010devi\u0107 dichotomy implies the Harrington\u2013Kechris\u2013Louveau dichotomy. We also give a simple proof of a graph-theoretic dichotomy of Miller for doubly-indexed sequences of analytic graphs, and show that this dichotomy generalizes to finite-dimensional hypergraphs but not to \u21350-dimensional hypergraphs.
\r\n\r\n5. An effective version of Nadkarni\u2019s Theorem was proved in Ditzen\u2019s unpublished Ph.D. thesis. The appendix contains a streamlined exposition of the proof and provides an alternative proof of the Effective Ergodic Decomposition Theorem for invariant measures (also originally proved by Ditzen). In addition, we show that the existence of an invariant Borel probability measure is not effective.
", "doi": "10.7907/w4nq-ag86", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:14608", "collection": "thesis", "collection_id": "14608", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05252022-040224796", "type": "thesis", "title": "Descriptive Set Theory and Dynamics of Countable Groups", "author": [ { "family_name": "Shinko", "given_name": "Forte", "orcid": "0000-0001-8142-1509", "clpid": "Shinko-Forte" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Conlon", "given_name": "David", "orcid": "0000-0001-5899-1829", "clpid": "Conlon-David" }, { "family_name": "Vidny\u00e1nszky", "given_name": "Zolt\u00e1n", "orcid": "0000-0001-8168-9353", "clpid": "Vidny\u00e1nszky-Zolt\u00e1n" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis comprises four papers.
\r\n\r\n1. We show that for any Polish group G and any countable normal subgroup \u0393 \u22b3 G, the coset equivalence relation G/\u0393 is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.
\r\n\r\n2. Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.
\r\n\r\n3. Let \u0393 be a countable group. A classical theorem of Thorisson states that if X is a standard Borel \u0393-space and \u00b5 and \u03bd are Borel probability measures on X which agree on every \u0393-invariant subset, then \u00b5 and \u03bd are equidecomposable,\r\ni.e., there are Borel measures (\u00b5\u03b3)\u03b3\u03f5\u0393 on X such that \u00b5 = \u03a3\u03b3\u00b5\u03b3 and \u03bd = \u03a3\u03b3\u03b3\u00b5\u03b3. We establish a generalization of this result to cardinal algebras.
\r\n\r\n4. Let R be a ring equipped with a proper norm. We show that under suitable conditions on R, there is a natural basis under continuous linear injection for the set of Polish R-modules which are not countably generated. When R is a division ring, this basis can be taken to be a singleton.
", "doi": "10.7907/egch-kp69", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:14206", "collection": "thesis", "collection_id": "14206", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05302021-185858708", "type": "thesis", "title": "Analogues of Amenability", "author": [ { "family_name": "Frisch", "given_name": "Joshua Robert", "clpid": "Frisch-Joshua-Ropbert" } ], "thesis_advisor": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Katz", "given_name": "Nets H.", "orcid": "0000-0002-6239-5429", "clpid": "Katz-N-H" }, { "family_name": "Vidny\u00e1nszky", "given_name": "Zolt\u00e1n", "orcid": "0000-0001-8168-9353", "clpid": "Vidny\u00e1nszky-Zolt\u00e1n" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis I study multiple notions related to and inspired by amenability coming from the points of the view of random walks on groups, dynamical systems, Borel equivalence relations, and descriptive linear algebra. In particular I study notions related to harmonic functions, invariant measures, hyperfiniteness, and dichotomy theorems.", "doi": "10.7907/yymw-xd55", "publication_date": "2021", "thesis_type": "phd", "thesis_year": "2021" }, { "id": "thesis:11006", "collection": "thesis", "collection_id": "11006", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012018-160828760", "primary_object_url": { "basename": "meehan_connor_2018.pdf", "content": "final", "filesize": 791323, "license": "other", "mime_type": "application/pdf", "url": "/11006/1/meehan_connor_2018.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Definable Combinatorics of Graphs and Equivalence Relations", "author": [ { "family_name": "Meehan", "given_name": "Connor George Walmsley", "orcid": "0000-0002-7596-2437", "clpid": "Meehan-Connor-George-Walmsley" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Lupini", "given_name": "Martino", "orcid": "0000-0003-1588-7057", "clpid": "Lupini-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let D = (X, D) be a Borel directed graph on a standard Borel space X and let \u03c7B(D) be its Borel chromatic number. If F0, \u2026, Fn-1: X \u2192 X are Borel functions, let DF0, \u2026, Fn-1 be the directed graph that they generate. It is an open problem if \u03c7B(DF0, \u2026, Fn-1) \u2208 {1, \u2026, 2n + 1, \u21350}. Palamourdas verified the foregoing for commuting functions with no fixed points. We show here that for commuting functions with the property that there is a path from each x \u2208 X to a fixed point of some Fj, there exists an increasing filtration X = \u22c3m < \u03c9 Xm such that \u03c7B(DF0, \u2026, Fn-1\u21be Xm) \u2264 2n for each m. We also prove that if n = 2 in the previous case, then \u03c7B(DF0, F1) \u2264 4. It follows that the approximate measure chromatic number \u03c7apM(D) \u2264 2n + 1 when the functions commute.
\r\n\r\nIf X is a set, E is an equivalence relation on X, and n \u2208 \u03c9, then define [X]nE = {(x0, ..., xn - 1) \u2208 nX: (\u2200i,j)(i \u2260 j \u2192 \u00ac(xi E xj))}. For n \u2208 \u03c9, a set X has the n-J\u00f3nsson property if and only if for every function f: [X]n= \u2192 X, there exists some Y \u2286 X with X and Y in bijection so that f[[Y]n=] \u2260 X. A set X has the J\u00f3nsson property if and only for every function f : (\u22c3n \u2208 \u03c9 [X]n=) \u2192 X, there exists some Y \u2286 X with X and Y in bijection so that f[\u22c3n \u2208 \u03c9 [Y]n=] \u2260 X. Let n \u2208 \u03c9, X be a Polish space, and E be an equivalence relation on X. E has the n-Mycielski property if and only if for all comeager C \u2286 nX, there is some Borel A \u2286 X so that E \u2264B E \u21be A and [A]nE \u2286 C. The following equivalence relations will be considered: E0 is defined on \u03c92 by x E0 y if and only if (\u2203n)(\u2200k > n)(x(k) = y(k)). E1 is defined on \u03c9(\u03c92) by x E1 y if and only if (\u2203n)(\u2200k > n)(x(k) = y(k)). E2 is defined on \u03c92 by x E2 y if and only if \u2211{1\u2044(n + 1): x(n) \u2260 y(n)} < \u221e. E3 is defined on \u03c9(\u03c92) by x E3 y if and only if (\u2200n)(x(n) E0 y(n)). Holshouser and Jackson have shown that \u211d is J\u00f3nsson under AD. The present research will show that E0 does not have the 3-Mycielski property and that E1, E2, and E3 do not have the 2-Mycielski property. Under ZF + AD, \u03c92/E0 does not have the 3-J\u00f3nsson property.
\r\n\r\nLet G = (X, G) be a graph and define for b \u2265 1 its b-fold chromatic number \u03c7(b)(G) as the minimum size of Y such that there is a function c from X into b-sets of Y with c(x) \u2229 c(y) = \u2205 if x G y. Then its fractional chromatic number is \u03c7f(G) = infb \u03c7(b)(G)\u2044b if the quotients are finite. If X is Polish and G is a Borel graph, we can also define its fractional Borel chromatic number \u03c7fB(G) by restricting to only Borel functions. We similarly define this for Baire measurable and \u03bc-measurable functions for a Borel measure \u03bc. We show that for each countable graph G, one may construct an acyclic Borel graph G' on a Polish space such that \u03c7fBM(G') = \u03c7f(G) and \u03c7BM(G') = \u03c7(G), and similarly for \u03c7f\u03bc and \u03c7\u03bc. We also prove that the implication \u03c7f(G) = 2 \u21d2 \u03c7(G) = 2 is false in the Borel setting.
", "doi": "10.7907/45E4-MC27", "publication_date": "2018", "thesis_type": "phd", "thesis_year": "2018" }, { "id": "thesis:10920", "collection": "thesis", "collection_id": "10920", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05212018-182542598", "primary_object_url": { "basename": "chen_ruiyuan_2018.pdf", "content": "final", "filesize": 1188145, "license": "other", "mime_type": "application/pdf", "url": "/10920/1/chen_ruiyuan_2018.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Definability and Classification of Equivalence Relations and Logical Theories", "author": [ { "family_name": "Chen", "given_name": "Ruiyuan", "orcid": "0000-0002-5891-8717", "clpid": "Chen-Ruiyuan" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Lupini", "given_name": "Martino", "orcid": "0000-0003-1588-7057", "clpid": "Lupini-M" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of four independent papers.
\r\n\r\nIn the first paper, joint with Kechris, we study the global aspects of structurability in the theory of countable Borel equivalence relations. For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We show that K-structurability interacts well with various preorders commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K; in particular, we characterize the K such that every K-structurable equivalence relation is smooth.
\r\n\r\nIn the second paper, we consider the classes of Kn-structurable equivalence relations, where Kn is the class of n-dimensional contractible simplicial complexes. We show that every Kn-structurable equivalence relation Borel embeds into one structurable by complexes in Kn with the further property that each vertex belongs to at most Mn := 2n-1(n2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n=1.
\r\n\r\nIn the third paper, we consider the amalgamation property from model theory in an abstract categorical context. A category is said to have the amalgamation property if every pushout diagram has a cocone. We characterize the finitely generated categories I such that in every category with the amalgamation property, every I-shaped diagram has a cocone.
\r\n\r\nIn the fourth paper, we prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic Lω1ω: every countable Lω1ω-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories (L,T) and (L',T'), every Borel functor Mod(L',T') → Mod(L,T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L'ω1ω-interpretation of T in T', which generalizes a recent result of Harrison-Trainor, Miller, and Montalban in the case where T, T' are ℵ0-categorical.
", "doi": "10.7907/7BP3-VZ93", "publication_date": "2018", "thesis_type": "phd", "thesis_year": "2018" }, { "id": "thesis:10236", "collection": "thesis", "collection_id": "10236", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05312017-155530848", "primary_object_url": { "basename": "chan_william_2017.pdf", "content": "final", "filesize": 735746, "license": "other", "mime_type": "application/pdf", "url": "/10236/1/chan_william_2017.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Aspects of Definability for Equivalence Relations", "author": [ { "family_name": "Chan", "given_name": "William", "orcid": "0000-0002-0661-1764", "clpid": "Chan-William" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Lupini", "given_name": "Martino", "orcid": "0000-0003-1588-7057", "clpid": "Lupini-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis will show that in the constructible universe L and set forcing extensions of L, there are no almost Borel reductions of the well-ordering equivalence relation into the admissibility equivalence relation and no Borel reductions of the isomorphism relation of any counterexample to Vaught's conjecture into the admissibility equivalence relation.
\r\n\r\nLet E be an analytic equivalence relation on a Polish space X with all classes Borel. Let I be a sigma-ideal on X such that its associated forcing of I-positive Borel subsets is a proper forcing. Assuming sharps of appropriate sets exist, it will be shown that there is an I-positive Borel subset of X on which the restriction of E is a Borel equivalence relation.
\r\n\r\nAssuming there are infinitely many Woodin cardinals below a measurable cardinal, then for any equivalence relation E in L(R) with all Borel classes and sigma-ideal I whose associated forcing is proper, there is an I-positive Borel set on which the restriction of E is Borel.
", "doi": "10.7907/Z90P0X3M", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:10213", "collection": "thesis", "collection_id": "10213", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05262017-150152046", "primary_object_url": { "basename": "approximation-classification-ergodic.pdf", "content": "final", "filesize": 621103, "license": "other", "mime_type": "application/pdf", "url": "/10213/1/approximation-classification-ergodic.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Approximation and Classification in the Ergodic Theory of Nonamenable Groups", "author": [ { "family_name": "Burton", "given_name": "Peter J.", "orcid": "0000-0002-0348-8385", "clpid": "Burton-Peter-J" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Katz", "given_name": "Nets H.", "orcid": "0000-0002-6239-5429", "clpid": "Katz-N-H" }, { "family_name": "Mantovan", "given_name": "Elena", "orcid": "0000-0003-4521-2130", "clpid": "Mantovan-E" }, { "family_name": "Lupini", "given_name": "Martino", "orcid": "0000-0003-1588-7057", "clpid": "Lupini-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis is a contribution to the theory of measurable actions of discrete groups on standard probability spaces. The focus is on nonamenable acting groups. It is organized into two parts. The first part deals with a notion called weak equivalence, which describes a sense in which such actions can approximate each other. The second part deals with the concept of entropy for measure preserving actions of sofic groups.
", "doi": "10.7907/Z9R49NTF", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:7716", "collection": "thesis", "collection_id": "7716", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05162013-102038765", "primary_object_url": { "basename": "Robin Tucker-Drob, Edited Thesis.pdf", "content": "final", "filesize": 1273374, "license": "other", "mime_type": "application/pdf", "url": "/7716/1/Robin Tucker-Drob, Edited Thesis.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Descriptive Set Theory and the Ergodic Theory of Countable Groups", "author": [ { "family_name": "Tucker-Drob", "given_name": "Robin Daniel", "clpid": "Tucker-Drob-Robin-Daniel" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Sokic", "given_name": "Miodrag", "clpid": "Sokic-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel reducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of residually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence. One of the main results answers a question of Ab\u00e9rt and Elek by showing that within any free weak equivalence class the isomorphism relation does not admit classification by countable structures. The proof relies on affirming a conjecture of Ioana by showing that the product of a free action with a Bernoulli shift is weakly equivalent to the original action. Chapter 5 studies the relationship between mixing and freeness properties of measure preserving actions. Chapter 6 studies how approximation properties of ergodic actions and unitary representations are reflected group theoretically and also operator algebraically via a group's reduced C*-algebra. Chapter 7 is an appendix which includes various results on mixing via filters and on Gaussian actions.", "doi": "10.7907/ER96-5D06", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:1584", "collection": "thesis", "collection_id": "1584", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05022008-113702", "primary_object_url": { "basename": "thesis-final.pdf", "content": "final", "filesize": 727712, "license": "other", "mime_type": "application/pdf", "url": "/1584/1/thesis-final.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Amenability, Countable Equivalence Relations, and Their Full Groups", "author": [ { "family_name": "Tsankov", "given_name": "Todor Dimitrov", "clpid": "Tsankov-Todor-Dimitrov" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Caicedo", "given_name": "Andr\u00e9s", "clpid": "Caicedo-A" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of an introduction and four independent chapters.
\r\n\r\nIn Chapter 1, we study homeomorphism groups of metrizable compactifications of the natural numbers. Those groups can be represented as almost zero-dimensional Polishable subgroups of the group S\u221e. We show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of S\u221e. We also find a sufficient condition for these groups to be one dimensional.
\r\n\r\nIn Chapter 2, we study the connections between properties of the action of a countable group \u0393 on a countable set X and the ergodic theoretic properties of the corresponding shift action of \u0393 on MX, where M is a measure space. In particular, we show that the action of \u0393 on X is amenable iff the corresponding shift has almost invariant sets. This is joint work with Alexander Kechris.
\r\n\r\nIn Chapter 3, we prove that if the Koopman representation associated to a measure-preserving action of a countable group on a standard non-atomic probability space is non-amenable, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an action on the boundary of a countably splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. This is joint work with Inessa Epstein.
\r\n\r\nIn Chapter 4, we study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of generators of a dense subgroup of full groups allowing us to distinguish full groups of equivalence relations generated by free, ergodic actions of the free groups Fn and Fm if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group iff its full group has a finitely generated dense subgroup. This is joint work with John Kittrell.\r\n
", "doi": "10.7907/N91C-HV48", "publication_date": "2008", "thesis_type": "phd", "thesis_year": "2008" }, { "id": "thesis:1896", "collection": "thesis", "collection_id": "1896", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05202005-160741", "primary_object_url": { "basename": "thesis.pdf", "content": "final", "filesize": 491373, "license": "other", "mime_type": "application/pdf", "url": "/1896/1/thesis.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Descriptive Properties of Measure Preserving Actions and the Associated Unitary Representations", "author": [ { "family_name": "Wei", "given_name": "Tzer-jen", "clpid": "Wei-Tzer-jen" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Wales", "given_name": "David B.", "clpid": "Wales-D-B" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of two independent parts.
\r\n\r\nIn the first part, we study the descriptive complexity of full groups [E]. The main result is
\r\n\r\n i) If E is not smooth, then [E] is Sigma\u2070\u20833 complete;
\r\n\r\n ii) If E is smooth, then [E] is closed.
In the second part, we study descriptive properties of the Koopman unitary repreesentation associated with the measure preserving action. We characterize the smoothness and compressibility of the equivalence induced by the unitary representaion. We also study many connections between the equivalence relation on L^2(X) and the equivalence relation on X.
", "doi": "10.7907/HECM-A179", "publication_date": "2005", "thesis_type": "phd", "thesis_year": "2005" }, { "id": "thesis:6783", "collection": "thesis", "collection_id": "6783", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01262012-103823681", "primary_object_url": { "basename": "Pavelich_j_2001.pdf", "content": "final", "filesize": 25118003, "license": "other", "mime_type": "application/pdf", "url": "/6783/1/Pavelich_j_2001.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Commuting Equivalence Relations and Scales on Differentiable Functions", "author": [ { "family_name": "Pavelich", "given_name": "Janet Mary", "clpid": "Pavelich-Janet-Mary" } ], "thesis_advisor": [ { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Clemens", "given_name": "John D.", "clpid": "Clemens-J-D" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "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", "doi": "10.7907/RHBN-ZN06", "publication_date": "2002", "thesis_type": "phd", "thesis_year": "2002" }, { "id": "thesis:10021", "collection": "thesis", "collection_id": "10021", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01232017-134252080", "primary_object_url": { "basename": "Sofronidis_NE_1999.pdf", "content": "final", "filesize": 23194987, "license": "other", "mime_type": "application/pdf", "url": "/10021/1/Sofronidis_NE_1999.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Topics in descriptive set theory related to equivalence relations, complex borel and analytic sets", "author": [ { "family_name": "Sofronidis", "given_name": "Nikolaos Efstathiou", "clpid": "Sofronidis-Nikolaos-Efstathiou" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" }, { "family_name": "Gao", "given_name": "S.", "clpid": "Gao-Sicun" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The purpose of this doctoral dissertation is first to show that certain kinds\r\nof invariants for measures, self-adjoint and unitary operators are as far from\r\ncomplete as possible and second to give new natural examples of complex\r\nBorel and analytic sets originating from Analysis and Geometry.
\r\n\r\nThe dissertation is divided in two parts.
\r\n\r\nIn the first part we prove that the measure equivalence relation and certain\r\nof its most characteristic subequivalence relations are generically S\u221e-\r\nergodic and unitary conjugacy of self-adjoint and unitary operators is generically\r\nturbulent.
\r\n\r\nIn the second part we prove that for any 0 ≤ \u03b1 < \u221e, the set of entire\r\nfunctions whose order is equal to \u03b1 is \u220f03-complete and the set of all sequences\r\nof entire functions whose orders converge to \u03b1 is \u220f05-complete. We also prove\r\nthat given any line in the plane and any cardinal number 1 ≤ n ≤ N0, the\r\nset of continuous paths in the plane tracing curves which admit at least n\r\ntangents parallel to the given line is \u03a311-complete and the set of differentiable\r\npaths of class C2 in the plane admitting a canonical parameter in [0,1] and\r\ntracing curves which have at least n vertices is also \u03a311-complete.
", "doi": "10.7907/vva8-1959", "publication_date": "1999", "thesis_type": "phd", "thesis_year": "1999" }, { "id": "thesis:10016", "collection": "thesis", "collection_id": "10016", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01202017-113604553", "primary_object_url": { "basename": "Li_X_1998.pdf", "content": "final", "filesize": 22161383, "license": "other", "mime_type": "application/pdf", "url": "/10016/1/Li_X_1998.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Some results on projective equivalence relations", "author": [ { "family_name": "Li", "given_name": "Xuhua", "clpid": "Li-Xuhua" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We construct a \u220f11 equivalence relation E on \u03c9\u03c9 for which there is no largest\r\nE-thin, E-invariant \u220f11 subset of \u03c9\u03c9. Then we lift our result to the general case.\r\nNamely, we show that there is a \u220f12n+1 equivalence relation for which there is\r\nno largest E-thin, E-invariant \u220f12n+1 set under projective determinacy. This\r\nanswers an open problem raised in Kechris [Ke2].
\r\n\r\nOur second result in this thesis is a representation for thin \u220f13 equivalence\r\nrelations on u\u03c9. Precisely, we show that for each thin \u220f13 equivalence relation\r\nE on u\u03c9, there is a \u039413 in the codes map p: \u03c9\u03c9 \u2192 u\u03c9 and a \u220f13 in the codes\r\nequivalence relation e on u\u03c9 such that for all real numbers x and y,
\r\n\r\nxEy \u2194 (p(x),p(y))\u2208 e
\r\n\r\nThis lifts Harrington's result about thin \u220f11 equivalence relations to thin \u220f13\r\nequivalence relations.
", "doi": "10.7907/pn2n-7z61", "publication_date": "1998", "thesis_type": "phd", "thesis_year": "1998" }, { "id": "thesis:6744", "collection": "thesis", "collection_id": "6744", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:12012011-093702619", "primary_object_url": { "basename": "Slawomir_sj_1995.pdf", "content": "final", "filesize": 29194331, "license": "other", "mime_type": "application/pdf", "url": "/6744/1/Slawomir_sj_1995.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Applications of descriptive set theory to topology and analysis", "author": [ { "family_name": "Solecki", "given_name": "Slawomir J.", "clpid": "Solecki-S-J" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In Chapter 1, we prove that for every family I of closed subsets of a Polish space each \u2211^1_1 set can be covered by countably many members of I or else contains a nonempty \u220f^0_2 set which cannot be covered by countably many members of I. We derive from it the general form of Hurewicz's theorem due to Kechris, Louveau, and Woodin, and a theorem of Feng on the open covering axiom. Also some well known theorems on finding \"big\" closed sets inside of \"big\" \u2211^1_1 sets are consequeces of our result. Chapter 2 consists of a joint work with A.S. Kechris. We prove that given a \u03c3-ideal I, the possibility of approximating each \u2211^1_1 set by a Borel set modulo I is equivalent to a definable form of the countable chain condition. This answers a question of Mauldin. We also characterize the meager ideal on a Polish group G as the only translation invariant, ccc \u03c3-ideal I on G such that each set from I is contained in an F_\u03c3 set from I. This partially verifies a conjecture of Kunen. In Chapter 3, we establish a theorem which gives sufficient conditions for a K_\u03c3 equivalence relation to continuously embed E_o. As a consequence of this result we show that no indecomposable continuum contains a Borel set which has precisely one point in common with each composant. This solves an old problem in the theory of continua. In Chapter 4, answering a question of A.S. Kechris, we prove that the Topological Vaught Conjecture holds for Polish groups admitting invariant metrics. We also answer a question of R.L.Sami by proving that there exist continuous actions of Polish abelian groups with non-Borel induced orbit eqivalence relations. Actually, we give a fully algebraic characterization of sequences of countable abelian groups (H_n) such that the group \u220f_n H_n has a continuous action with non-Borel orbit equivalence relation. In Chapter 5, we give a characterization of local compactness for Polish abelian groups in terms of Haar null sets of Christensen: a Polish abelian group is locally compact if each family of mutually disjoint closed (or, equivalently, universally measurable) sets which are not Haar null is countable. This completes, in a sense Dougherty's solution to a problem of Christensen. We also consider the question of the possibility of approximating analytic by Borel sets modulo Haar null sets. Chapter 6 contains two dichotomy theorems for Baire class 1 functions: a Baire class 1 function can be decomposed into countably many continuous functions, or else it contains a function which is as complicated with respect to decompositions into continuous functions as any other Baire class 1 function; an analogous theorem is proved for decompositions into continuous functions with closed domains. These results strengthen a theorem of Jayne and Rogers and answer some questions of Steprans. Their proofs use effective descriptive set theory as well as infinite games. Some results on decompositions of Borel sets and functions on higher levels are also obtained.
\r\n", "doi": "10.7907/s2fn-6633", "publication_date": "1995", "thesis_type": "phd", "thesis_year": "1995" }, { "id": "thesis:4040", "collection": "thesis", "collection_id": "4040", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-10112007-111738", "type": "thesis", "title": "Topics in descriptive set theory related to number theory and analysis", "author": [ { "family_name": "Ki", "given_name": "Haseo", "clpid": "Ki-Haseo" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Based on the point of view of descriptive set theory, we have investigated several definable sets from number theory and analysis.
\r\n\r\nIn Chapter 1 we solve two problems due to Kechris about sets arising in number theory, provide an example of a somewhat natural D2\u03a003 set, and exhibit an exact relationship between the Borel class of a nonempty subset X of the unit interval and the class of subsets of N whose densities lie in X.
\r\n\r\nIn Chapter 2 we study the A, S, T and U-sets from Mahler's classification of complex numbers. We are able to prove that U and T are \u03a303-complete and \u03a003-complete respectively. In particular, U provides a rare example of a natural \u03a303-complete set.
\r\n\r\nIn Chapter 3 we solve a question due to Kechris about UCF, the set of all continuous functions, on the unit circle, with Fourier series uniformly convergent. We further show that any \u03a303 set, which contains UCF, must contain a continuous function with Fourier series divergent.
\r\n\r\nIn Chapter 4 we use techniques from number theory and the theory of Borel equivalence relations to provide a class of complete \u03a011 sets.
\r\n\r\nFinally, in Chapter 5, we solve a problem due to Ajtai and Kechris. For each differentiable function f on the unit circle, the Kechris-Woodin rank measures the failure of continuity of the derivative function f', while the Zalcwasser rank measures how close the Fourier series of f is to being a uniformly convergent series. We show that the Kechris-Woodin rank is finer than the Zalcwasser rank.
\r\n", "doi": "10.7907/SQV8-S991", "publication_date": "1995", "thesis_type": "phd", "thesis_year": "1995" }, { "id": "thesis:6737", "collection": "thesis", "collection_id": "6737", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:11232011-134429806", "primary_object_url": { "basename": "Ditzen_a_1992.pdf", "content": "final", "filesize": 2458721, "license": "other", "mime_type": "application/pdf", "url": "/6737/1/Ditzen_a_1992.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Definable equivalence relations on Polish spaces", "author": [ { "family_name": "Ditzen", "given_name": "Achim", "clpid": "Ditzen-A" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In the first part of this work we deal with the classification of definable equivalence relations on Polish spaces, where we take definable to mean inside some model of determinacy: We work in ZF+DC+AD_R. The classification is up to bireducebility (denoted by E~F), that is if E and F are equivalence relations on the Baire space N, then E ~F, if there is a mapping f : N \u2192 N with \u2200x, y \u2208 N (xEy \u21d4 f(x)Ff(y)), called a reduction of E into F, and vice versa.
\r\n\r\nAs two equivalence relations on Polish spaces are bireducible just in case there is a bijection between their quotient spaces, our results apply to de-finable cardinality theory, too. We show that up to bireducibility there are only four infinite hypersmooth equivalence relations: equality on the integers, equality on the Baire space, E_o on the Cantor space 2^\u03c9 given by\r\n\u03b1E_o\u03b2 \u21d4 \u2203n \u2208 w\u2200m > n (a(m) = ,\u03b2(m)),\r\nand E_1 on the countable product of Cantor space (2^\u03c9)^\u03c9 given by \u03b1E_0^\u03b2 \u21d4 \u2203n \u2208 w\u2200m > n (a_m = \u03b2_m).
\r\n\r\nEven though we only develop the theory for the context of ADR, it is clear from the proofs that our results apply to a variety of other settings, such as the one encountered in the second part.
\r\n\r\nIn the second part of the thesis we deal with countable Borel equivalence relations E on Polish spaces X, that is with equivalence relations which have countable classes and Borel graphs. The space M of probability measures on these spaces is again Polish. Of special interest are invariant measures (i.e. those which are preserved under bijections f : X \u2192 X with f(x)Ex, so called automorphisms), quasiinvariant measures (i.e. those whose measure class is preserved under automorphisms), and ergodic measures (i.e. those which assign full or null measure to E-invariant Borel sets).
\r\n\r\nWe show that the collections of ergodic measures and of ergodic quasiin-variant measures are Borel. We also classify the complexity of the \u03c3-ideal of nullsets with respect to all invariant measures, showing that this ideal is \u041f^1_1 in the codes of \u2206^l_1 and \u2211^l_1 sets, and that the \u03c3-ideal of compact nullsets with respect to all invariant measures is \u041f^0_2 if the collection of invariant ergodic measures is at most countable, and \u041f^1_1 -complete otherwise.
", "doi": "10.7907/rf5s-m868", "publication_date": "1992", "thesis_type": "phd", "thesis_year": "1992" }, { "id": "thesis:8783", "collection": "thesis", "collection_id": "8783", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:03182015-110250011", "type": "thesis", "title": "Smooth sets for borel equivalence relations and the covering property for \u03c3-ideals of compact sets", "author": [ { "family_name": "Uzc\u00e1tegui", "given_name": "Carlos", "clpid": "Uzc\u00e1tegui-C" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis is divided into three chapters. In the first chapter we study\r\nthe smooth sets with respect to a Borel equivalence realtion E on a Polish\r\nspace X. The collection of smooth sets forms \u03c3-ideal. We think of smooth\r\nsets as analogs of countable sets and we show that an analog of the perfect\r\nset theorem for \u03a311 sets holds in the context of smooth sets. We also show\r\nthat the collection of \u03a311 smooth sets is \u220f11 on the codes. The analogs of\r\nthin sets are called sparse sets. We prove that there is a largest \u220f11 sparse set\r\nand we give a characterization of it. We show that in L there is a \u220f11 sparse\r\nset which is not smooth. These results are analogs of the results known for\r\nthe ideal of countable sets, but it remains open to determine if large cardinal\r\naxioms imply that \u220f11 sparse sets are smooth. Some more specific results are\r\nproved for the case of a countable Borel equivalence relation. We also study\r\nI(E), the \u03c3-ideal of closed E-smooth sets. Among other things we prove that\r\nE is smooth iff I(E) is Borel.
\r\n\r\nIn chapter 2 we study \u03c3-ideals of compact sets. We are interested in the\r\nrelationship between some descriptive set theoretic properties like thinness,\r\nstrong calibration and the covering property. We also study products of \u03c3-ideals\r\nfrom the same point of view. In chapter 3 we show that if a \u03c3-ideal\r\nI has the covering property (which is an abstract version of the perfect set\r\ntheorem for \u03a311 sets), then there is a largest \u220f11 set in Iint (i.e., every closed\r\nsubset of it is in I). For \u03c3-ideals on 2\u03c9 we present a characterization of this\r\nset in a similar way as for C1, the largest thin \u220f11 set. As a corollary we get\r\nthat if there are only countable many reals in L, then the covering property\r\nholds for \u03a312 sets.
", "doi": "10.7907/cfsq-aw69", "publication_date": "1990", "thesis_type": "phd", "thesis_year": "1990" }, { "id": "thesis:2572", "collection": "thesis", "collection_id": "2572", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-06132007-073731", "primary_object_url": { "basename": "Kafkoulis_g_1990.pdf", "content": "final", "filesize": 3253446, "license": "other", "mime_type": "application/pdf", "url": "/2572/1/Kafkoulis_g_1990.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Homogeneous sequences of cardinals for ordinal definable partition relations", "author": [ { "family_name": "Kafkoulis", "given_name": "George", "clpid": "Kafkoulis-G" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.\n\nIn this dissertation we study the consistency strength of the theory ZFC & ([...] strong limit)([...] < [...])([...]) (*), and we prove the consistency of this theory relative to the consistency of the existence of a supercompact cardinal and an inaccessible above it. If U is a normal measure on [...], then [...] denotes the Supercompact Prikry forcing induced by U. [...] is the partition relation [...] except that we consider only OD colorings of [...]. Theorems 1,2 are the main results of our thesis.\n\nTheorem 1. If there exists a model of ZFC in which [...] is a supercompact cardinal and [...] is an innaccessible above [...], then we can construct a model V of the same properties with the additional property that if U is a normal [...]-measure and G is [...] - generic over V, then V[G] does not satisfy the [...] partition property. [...]\n\nIf G is a [...]-generic over V filter, then we define H to be the set H: [...], and we consider the inner model V(H), which is the smallest inner model of ZF that contains H as an element. We prove that V(H) satisfies the above partition property (*).\n\nMoreover, V(H) satisfies < [...] - DC and using this fact we define a forcing [...], which is almost-homogeneous, < [...] - closed forcing that forces the AC over V(H) and does not add any new sets of rank < [...].\n\nTheorem 2. If [...] is [...] -generic over V(H) and V[...], then [...] + [...] strong limit + [...]. Therefore Con(ZFC + ([...]) [...] supercompact & [...] inaccessible & is [...]) [...] Con(ZFC + ([...] strong limit)[...]. [...]\n", "doi": "10.7907/hchw-ca34", "publication_date": "1990", "thesis_type": "phd", "thesis_year": "1990" }, { "id": "thesis:1988", "collection": "thesis", "collection_id": "1988", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05232007-115904", "type": "thesis", "title": "Geodesic Flows on Manifolds of Negative Curvature with Smooth Horospheric Foliations", "author": [ { "family_name": "Feres", "given_name": "Renato", "clpid": "Feres-Renato" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Katok", "given_name": "Anatole", "clpid": "Katok-A" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We improve a result due to M. Kanai on the rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable (horospheric) foliation is smooth. More precisely, the main result proven here is: Let M be a closed C\u221e Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow \u03c6t: V \u2192 V on the unit tangent bundle V of M is C\u221e. Assume moreover that either (a) the sectional curvature of M satisfies -4 < K \u2264 -1 or (b) the dimension of M is odd. Then the geodesic flow of M is C\u221e-isomorphic (i. e., conjugate under a C\u221e diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature.
", "doi": "10.7907/f6yt-bf73", "publication_date": "1989", "thesis_type": "phd", "thesis_year": "1989" }, { "id": "thesis:6854", "collection": "thesis", "collection_id": "6854", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:03192012-101000956", "type": "thesis", "title": "Resource-Bounded Category and Measure in Exponential Complexity Classes", "author": [ { "family_name": "Lutz", "given_name": "Jack Harold", "orcid": "0000-0003-1004-3891", "clpid": "Lutz-Jack-Harold" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Abu-Mostafa", "given_name": "Yaser S.", "orcid": "0009-0005-1380-8345", "clpid": "Abu-Mostafa-Y-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis presents resource-bounded category and resource bounded-measure - two new tools for computational complexity theory - and some applications of these tools to the structure theory of exponential complexity classes.
\r\n\r\nResource-bounded category, a complexity-theoretic version of the classical Baire category method, identifies certain subsets of PSPACE, E, ESPACE, and other complexity classes as meager. These meager sets are shown to form a nontrivial ideal of \"small\" subsets of the complexity class. The meager sets are also (almost) characterized in terms of certain two-person infinite games called resource-bounded Banach-Mazur games.
\r\n\r\nSimilarly, resource-bounded measure, a complexity-theoretic version of Lebesgue measure theory, identifies the measure 0 subsets of E, ESPACE, and other complexity classes, and these too are shown to form nontrivial ideals of \"small\" subsets. A resource-bounded extension of the classical Kolmogorov zero-one law is also proven. This shows that measurable sets of complexity-theoretic interest either have measure 0 or are the complements of sets of measure 0.
\r\n\r\nResource-bounded category and measure are then applied to the investigation of uniform versus nonuniform complexity. In particular, Kannan's theorem that ESPACE \u228a P/Poly is extended by showing that P/ Poly \u2229 ESPACE is only a meager, measure 0 subset of ESPACE. A theorem of Huynh is extended similarly by showing that all but a meager, measure 0 subset of the languages in ESPACE have high space-bounded Kolmogorov complexity.
\r\n\r\nThese tools are also combined with a new hierarchy of exponential time complexity classes to refine known relationships between nonuniform complexity and time complexity.
\r\n\r\nIn the last part of the thesis, known properties of hard languages are extended. In particular, recent results of Sch\u00f6ning and Huynh state that any language L which is \u2264Pm-hard for E or \u2264PT-hard for ESPACE cannot be feasibly approximated (i.e., its symmetric difference with any feasible language has exponential density). It is proven here that this conclusion in fact holds unless only a meager subset of E is \u2264Pm-reducible to L and only a meager, measure 0 subset of ESPACE is \u2264PSPACEm-reducible to L. (It is conjectured, but not proven, that this result is actually stronger than those of Sch\u00f6ning and Huynh.) This suggests a new lower bound method which may be useful in interesting cases.
", "doi": "10.7907/dccw-3z56", "publication_date": "1987", "thesis_type": "phd", "thesis_year": "1987" }, { "id": "thesis:10017", "collection": "thesis", "collection_id": "10017", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01202017-145315777", "primary_object_url": { "basename": "Ramsamujh_TI_1986.pdf", "content": "final", "filesize": 36155563, "license": "other", "mime_type": "application/pdf", "url": "/10017/1/Ramsamujh_TI_1986.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Some Topics in Descriptive Set Theory and Analysis", "author": [ { "family_name": "Ramsamujh", "given_name": "Taje Indrallal", "clpid": "Ramsamujh-Taje-Indrallal" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" }, { "family_name": "Wolff", "given_name": "Thomas H.", "clpid": "Wolff-T-H" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "De Prima", "given_name": "Charles R.", "clpid": "De-Prima-C-R" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Coanalytic subsets of some well known Polish spaces are investigated. A natural norm (rank function) on each subset is defined and studied by using well-founded trees and transfinite induction as the main tools. The norm provides a natural measure of the complexity of the elements in each subset. It also provides a \"Rank Argument\" of the non-Borelness of the subset.
\r\n\r\nThe work is divided into four chapters. In Chapter 1 nowhere differentiable continuous functions and Besicovitch functions are studied. Chapter 2 deals with functions with everywhere divergent Fourier series, and everywhere divergent trigonometric series with coefficients that tend to zero. Compact Jordan sets (i.e., sets without cavities) and compact simply-connected sets in the plane are investigated in Chapter 3. Chapter 4 is a miscellany of results extending earlier work of M. Ajtai, A. Kechris and H. Woodin on differentiable functions and continuous functions with everywhere convergent Fourier series.
", "doi": "10.7907/8pdn-xf41", "publication_date": "1986", "thesis_type": "phd", "thesis_year": "1986" }, { "id": "thesis:10020", "collection": "thesis", "collection_id": "10020", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01232017-115539334", "primary_object_url": { "basename": "Crawshaw_M_1985.pdf", "content": "final", "filesize": 10835351, "license": "other", "mime_type": "application/pdf", "url": "/10020/1/Crawshaw_M_1985.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Explicit Formulas for the Jump of Q-Degrees", "author": [ { "family_name": "Crawshaw", "given_name": "Mark", "clpid": "Crawshaw-Mark" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Becker", "given_name": "Howard Starr", "clpid": "Becker-Howard-Starr" }, { "family_name": "Lorden", "given_name": "Gary A.", "clpid": "Lorden-G-A" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Wales", "given_name": "David B.", "clpid": "Wales-D-B" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In the context of the axiom of projective determinacy, Q-degrees have been proposed as the appropriate generalisations of the hyperdegrees to all the odd levels of the projective hierarchy. In chapter one we briefly review the basics of Q-theory.
\r\n\r\nIn the second chapter we characterise the Q-jump operation in terms of certain two-person games and derive an explicit formula for the Q-jump. This makes clear the similarities between the Q-degrees and the constructibility degrees, the Q-jump operation being a natural generalisation of the sharp operation.
\r\n\r\nIn chapter three we mix our earlier results with some\r\nforcing techniques to get a new proof of the jump inversion theorem for Q-degrees. We also extend some results about minimal covers in hyperdegrees to the Q-degrees. Many of our methods are immediately applicable to the constructible degrees and provide new proofs of old results.
", "doi": "10.7907/b0x4-c027", "publication_date": "1985", "thesis_type": "phd", "thesis_year": "1985" }, { "id": "thesis:10018", "collection": "thesis", "collection_id": "10018", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01202017-152612696", "primary_object_url": { "basename": "Kastanas_IG_1981.pdf", "content": "final", "filesize": 13689344, "license": "other", "mime_type": "application/pdf", "url": "/10018/1/Kastanas_IG_1981.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "The Ramsey Property and Degrees in the Analytical Hierarchy", "author": [ { "family_name": "Kastanas", "given_name": "Ilias George", "clpid": "Kastanas-Ilias-George" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In Chapter I we review some known results about the Ramsey \r\ntheory for partitions of reals, and we present a certain two-person \r\ngame such that if either player has a winning strategy then a homogeneous \r\nset for the partition can be constructed, and conversely. \r\nThis gives alternative proofs of some of the known results. We \r\nthen discuss possible uses of the game in obtaining effective\r\nversions and prove a theorem along these lines.
\r\n\r\nIn Chapter II we study the structure of initial segments of the\r\n\u039412n+1-degrees, assuming Projective Determinacy. We show that every \r\nfinite distributive lattice is isomorphic to such an initial segment, \r\nand hence that the first-order theory of the ordering of \u039412n+1-degrees \r\nis undecidable.
\r\n\r\nIn Chapter III we extend Friedberg's Jump Inversion theorem to \r\nQ2n+1-degrees, after noticing that it fails tor \u039412n+1-degrees. We\r\nassume again Projective Determinacy.
", "doi": "10.7907/pp79-pm74", "publication_date": "1981", "thesis_type": "phd", "thesis_year": "1981" }, { "id": "thesis:8901", "collection": "thesis", "collection_id": "8901", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05282015-144505104", "primary_object_url": { "basename": "Rubin_a_1978.pdf", "content": "final", "filesize": 5132528, "license": "other", "mime_type": "application/pdf", "url": "/8901/1/Rubin_a_1978.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Free Algebras in Von Neumann-Bernays-G\u04e7del Set Theory and Positive Elementary Inductions in Reasonable Structures", "author": [ { "family_name": "Rubin", "given_name": "Arthur Leonard", "clpid": "Arthur Leonard" } ], "thesis_advisor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of two independent chapters. The first chapter deals with universal algebra. It is shown, in von Neumann-Bernays-G\u04e7del set theory, that free images of partial algebras exist in arbitrary varieties. It follows from this, as set-complete Boolean algebras form a variety, that there exist free set-complete Boolean algebras on any class of generators. This appears to contradict a well-known result of A. Hales and H. Gaifman, stating that there is no complete Boolean algebra on any infinite set of generators. However, it does not, as the algebras constructed in this chapter are allowed to be proper classes. The second chapter deals with positive elementary inductions. It is shown that, in any reasonable structure \u1d86, the inductive closure ordinal of \u1d86 is admissible, by showing it is equal to an ordinal measuring the saturation of \u1d86. This is also used to show that non-recursively saturated models of the theories ACF, RCF, and DCF have inductive closure ordinals greater than \u03c9.", "doi": "10.7907/62t8-9b85", "publication_date": "1978", "thesis_type": "phd", "thesis_year": "1978" } ]