[ { "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\n

In 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\n

In 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\n

In 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\n

In 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: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\n

In 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\n

In 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\n

In 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\n

In 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: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\n

If 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\n

Let 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: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\n

If 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\n

Let 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: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\n

Let 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\n

Assuming 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: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\n

Let 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\n

Assuming 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" } ]