{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98378"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98378","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Structures and dynamics","abstract":"\"Our results are divided in three independent chapters. In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same applies to the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: \"\"For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M).\"\" From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15]. In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16]. In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16].\"","abstract_html":""Our results are divided in three independent chapters. In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same applies to the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: ""For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M)."" From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15]. In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16]. In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16]."","abstract_has_math":false,"creators":["Panagiotopoulos, Aristotelis"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Solecki, Slawek","Hieronymi, Philipp","Tserunyan, Anush","van den Dries, Lou"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T17:56:45Z","date_published":"2017-09-29T17:56:45Z","updated_at":"2026-07-22T22:24:35Z","subjects":["Polish groups","Fraisse","Turbulence","Hjorth","Left invariant","Becker","Projective Fraisse","Infinite games","Borel complexity"],"languages":["en"],"rights":["Copyright 2017 Aristotelis Panagiotopoulos"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98378","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Solecki, Slawek","Hieronymi, Philipp","Tserunyan, Anush","van den Dries, Lou"]},{"key":"dc:creator","label":"Author","values":["Panagiotopoulos, Aristotelis"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-09-29T17:56:45Z","2017-07-13","2017-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Polish groups","Fraisse","Turbulence","Hjorth","Left invariant","Becker","Projective Fraisse","Infinite games","Borel complexity"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Aristotelis Panagiotopoulos"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98378"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"Our results are divided in three independent chapters. In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same applies to the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: \"\"For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M).\"\" From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15]. In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16]. In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16].\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Aristotelis Panagiotopoulos, accepted the attached license on 2017-07-12 at 12:09.","The student, Aristotelis Panagiotopoulos, submitted this Dissertation for approval on 2017-07-12 at 12:49.","This Dissertation was approved for publication on 2017-07-13 at 17:51.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11424 on 2017-09-29 at 11:29:54","Made available in DSpace on 2017-09-29T17:56:45Z (GMT). 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Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: \"\"For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M).\"\" From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15]. In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16]. In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16].\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Aristotelis Panagiotopoulos, accepted the attached license on 2017-07-12 at 12:09.","The student, Aristotelis Panagiotopoulos, submitted this Dissertation for approval on 2017-07-12 at 12:49.","This Dissertation was approved for publication on 2017-07-13 at 17:51.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11424 on 2017-09-29 at 11:29:54","Made available in DSpace on 2017-09-29T17:56:45Z (GMT). No. of bitstreams: 3 PANAGIOTOPOULOS-DISSERTATION-2017.pdf: 698055 bytes, checksum: fd2145a4b9d7125ce516207a5c37a371 (MD5) LICENSE.txt: 4224 bytes, checksum: dc5ab09e9cbbf20e57fa183bef5b82af (MD5) PROQUEST_LICENSE.txt: 4570 bytes, checksum: 378bcaca289f6a60e9cee495d112d1d1 (MD5) Previous issue date: 2017-07-13"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/98378"],"dc:language":["en"],"dc:rights":["Copyright 2017 Aristotelis Panagiotopoulos"],"dc:subject":["Polish groups","Fraisse","Turbulence","Hjorth","Left invariant","Becker","Projective Fraisse","Infinite games","Borel complexity"],"dc:title":["Structures and dynamics"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:35Z"}