[ { "id": "authors:ywbjj-n1p25", "collection": "authors", "collection_id": "ywbjj-n1p25", "cite_using_url": "https://authors.library.caltech.edu/records/ywbjj-n1p25", "type": "article", "title": "Borel graphable equivalence relations", "author": [ { "family_name": "Arant", "given_name": "Tyler" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Lutz", "given_name": "Patrick" } ], "abstract": "

This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible ordinals and show that it is Borel graphable if and only if there is a non-constructible real. As a corollary of the proof, we construct an analytic equivalence relation which is (provably in ZFC) not Borel graphable and an effectively analytic equivalence relation which is Borel graphable but not effectively Borel graphable. Next, we study analytic equivalence relations given by the isomorphism relation for some class of countable structures. We show that all such equivalence relations are Borel graphable, which implies that for every Borel action of S ∞, the associated orbit equivalence relation is Borel graphable. This leads us to study the class of Polish groups whose Borel actions always give rise to Borel graphable orbit equivalence relations; we refer to such groups as graphic groups. We show that besides S ∞, the class of graphic groups includes all connected Polish groups and is closed under countable products. We finish by studying structural properties of the class of Borel graphable analytic equivalence relations and by considering two variations on Borel graphability: a generalization with hypergraphs instead of graphs and an analogue of Borel graphability in the setting of computably enumerable equivalence relations.

", "doi": "10.1016/j.aim.2025.110765", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "2026-03", "volume": "487", "pages": "110765" }, { "id": "authors:25tmg-55397", "collection": "authors", "collection_id": "25tmg-55397", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230125-514112400.2", "type": "article", "title": "Lifts of Borel actions on quotient spaces", "author": [ { "family_name": "Frisch", "given_name": "Joshua R.", "clpid": "Frisch-Joshua-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Shinko", "given_name": "Forte", "orcid": "0000-0001-8142-1509", "clpid": "Shinko-Forte" } ], "abstract": "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.", "doi": "10.1007/s11856-022-2434-z", "issn": "0021-2172", "publisher": "Springer", "publication": "Israel Journal of Mathematics", "publication_date": "2022-12", "series_number": "2", "volume": "251", "issue": "2", "pages": "379-421" }, { "id": "authors:yk4b8-2q134", "collection": "authors", "collection_id": "yk4b8-2q134", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220107-488030400", "type": "article", "title": "On Polish groups admitting non-essentially countable actions", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Malicki", "given_name": "Maciej", "clpid": "Malicki-Maciej" }, { "family_name": "Panagiotopoulos", "given_name": "Aristotelis", "orcid": "0000-0002-7695-4842", "clpid": "Panagiotopoulos-Aristotelis" }, { "family_name": "Zielinski", "given_name": "Joseph", "clpid": "Zielinski-Joseph" } ], "abstract": "It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.", "doi": "10.1017/etds.2020.133", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "2022-01", "series_number": "1", "volume": "42", "issue": "1", "pages": "180-194" }, { "id": "authors:gdfsz-5f343", "collection": "authors", "collection_id": "gdfsz-5f343", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-090422831", "type": "article", "title": "Weak containment of measure-preserving group actions", "author": [ { "family_name": "Burton", "given_name": "Peter J.", "clpid": "Burton-P-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.", "doi": "10.1017/etds.2019.26", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "2020-10", "series_number": "10", "volume": "40", "issue": "10", "pages": "2681-2733" }, { "id": "authors:pxrsq-vyd35", "collection": "authors", "collection_id": "pxrsq-vyd35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191205-100802061", "type": "article", "title": "Co-induction and invariant random subgroups", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Quorning", "given_name": "Vibeke", "clpid": "Quorning-V" } ], "abstract": "In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group.\nWe use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of F\u2082 which are all invariant and weakly mixing with respect to the action of Aut(F\u2082).\nMoreover, for amenable groups \u0393 \u2264 \u0394, we obtain that the standard co-induction operation from the space of weak equivalence classes of \u0393\u0393 to the space of weak equivalence classes of \u0394 is continuous if and only if [\u0394:\u0393] < \u221e or core \u0394(\u0393) is trivial. For general groups we obtain that the co-induction operation is not continuous when [\u0394:\u0393] = \u221e. This answers a question raised by Burton and Kechris in [17]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [8].", "doi": "10.4171/GGD/517", "issn": "1661-7207", "publisher": "European Mathematical Society", "publication": "Groups, Geometry, and Dynamics", "publication_date": "2019-05-07", "series_number": "4", "volume": "13", "issue": "4", "pages": "1151-1193" }, { "id": "authors:ah2hj-4fp82", "collection": "authors", "collection_id": "ah2hj-4fp82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-083759861", "type": "article", "title": "The complexity of topological group isomorphism", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Nies", "given_name": "Andr\u00e9", "clpid": "Nies-A" }, { "family_name": "Tent", "given_name": "Katrin", "clpid": "Tent-K" } ], "abstract": "We study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.", "doi": "10.1017/jsl.2018.25", "issn": "0022-4812", "publisher": "Cambridge University Press", "publication": "Journal of Symbolic Logic", "publication_date": "2018-09", "series_number": "3", "volume": "83", "issue": "3", "pages": "1190-1203" }, { "id": "authors:gsa52-xsw68", "collection": "authors", "collection_id": "gsa52-xsw68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-102415675", "type": "article", "title": "Structurable equivalence relations", "author": [ { "family_name": "Chen", "given_name": "Ruiyuan", "clpid": "Chen-Ruiyuan" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "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 study in this paper the global structure of the classes of K-structurable equivalence relations for various K. We show that K-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in\nthe classification of countable Borel equivalence relations. We consider the poset of classes\nof K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice; this implies that the Borel reducibility preordering among countable Borel equivalence relations contains a large sublattice. 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, answering a question of Marks.", "doi": "10.4064/fm428-7-2017", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "2018-03-01", "volume": "242", "pages": "109-185" }, { "id": "authors:yzec3-5ya52", "collection": "authors", "collection_id": "yzec3-5ya52", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-091809776", "type": "article", "title": "Invariant random subgroups and action versus representation maximality", "author": [ { "family_name": "Burton", "given_name": "Peter J.", "clpid": "Burton-P-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We show that weak containment of free ergodic measure-preserving actions of F\u221e is not equivalent to weak containment of the corresponding Koopman representations. This result is based on the construction of an invariant random subgroup of F\u221e which is supported on the maximal actions.", "doi": "10.1090/proc/13591", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "2017-09", "series_number": "9", "volume": "145", "issue": "9", "pages": "3961-3971" }, { "id": "authors:9sngr-rxs67", "collection": "authors", "collection_id": "9sngr-rxs67", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161130-102008360", "type": "article", "title": "Borel equivalence relations and cardinal algebras", "author": [ { "family_name": "Macdonald", "given_name": "Henry L.", "clpid": "Macdonald-H-L" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We show that Tarski's concept of cardinal algebra appears naturally in the context of the current theory of Borel equivalence relations. As a result one can apply Tarski's theory to discover a number of interesting laws governing the structure of Borel equivalence relations, which, in retrospect rather surprisingly, have not been realized before.", "doi": "10.4064/fm242-4-2016", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "2016-06-17", "volume": "235", "pages": "183-198" }, { "id": "authors:jeks1-rav12", "collection": "authors", "collection_id": "jeks1-rav12", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150107-075104911", "type": "article", "title": "Random orderings and unique ergodicity of automorphism groups", "author": [ { "family_name": "Angel", "given_name": "Omer", "clpid": "Angel-O" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Lyons", "given_name": "Russell", "clpid": "Lyons-R" } ], "abstract": "We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner and Weiss's example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of graphs and metric spaces in which such random orderings can be found that are not uniform.", "doi": "10.4171/JEMS/483", "issn": "1435-9855", "publisher": "European Mathematical Society", "publication": "Journal of the European Mathematical Society", "publication_date": "2014", "series_number": "10", "volume": "16", "issue": "10", "pages": "2059-2095" }, { "id": "authors:fsang-0q918", "collection": "authors", "collection_id": "fsang-0q918", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20131216-115036936", "type": "article", "title": "Stationary probability measures and topological realizations", "author": [ { "family_name": "Conley", "given_name": "Clinton T.", "clpid": "Conley-C-T" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Miller", "given_name": "Benjamin D.", "clpid": "Miller-B-D" } ], "abstract": "We establish the generic inexistence of stationary Borel probability measures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no \u03c3-compact realization.", "doi": "10.1007/s11856-013-0025-8", "issn": "0021-2172", "publisher": "Springer", "publication": "Israel Journal of Mathematics", "publication_date": "2013-11", "series_number": "1", "volume": "198", "issue": "1", "pages": "333-345" }, { "id": "authors:22xvc-qcf35", "collection": "authors", "collection_id": "22xvc-qcf35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130405-083026269", "type": "article", "title": "Ultraproducts of measure preserving actions and graph combinatorics", "author": [ { "family_name": "Conley", "given_name": "Clinton T.", "clpid": "Conley-C-T" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tucker-Drob", "given_name": "Robin D.", "clpid": "Tucker-Drob-R-D" } ], "abstract": "Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations.", "doi": "10.1017/S0143385711001143", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "2013-04", "volume": "33", "pages": "334-374" }, { "id": "authors:5fpz2-tn671", "collection": "authors", "collection_id": "5fpz2-tn671", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-131021822", "type": "article", "title": "Measurable chromatic and independence numbers for ergodic\n graphs and group actions", "author": [ { "family_name": "Conley", "given_name": "Clinton T.", "clpid": "Conley-C-T" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study in this paper combinatorial problems concerning graphs generated by measure preserving actions of countable groups on standard measure spaces. In particular we study chromatic and independence numbers, in both the measure-theoretic and the Borel context, and relate the behavior of these parameters to properties of the acting group such as amenability, Kazhdan's property (T), and freeness. We also prove a Borel analog of the classical Brooks' Theorem in finite combinatorics for actions of groups with finitely many ends.", "doi": "10.4171/GGD/179", "issn": "1661-7207", "publisher": "European Mathematical Society", "publication": "Groups, Geometry, and Dynamics", "publication_date": "2013", "series_number": "1", "volume": "7", "issue": "1", "pages": "127-180" }, { "id": "authors:0k1qp-a0g45", "collection": "authors", "collection_id": "0k1qp-a0g45", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120716-102713726", "type": "article", "title": "Weak containment in the space of actions of a free group", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is shown that the translation action of the free group with n generators\non its profinite completion is the maximum, in the sense of weak containment,\nmeasure preserving action of this group. Using also a result of\nAb\u00e9rt\u2013Nikolov this is used to give a new proof of Gaboriau's theorem that\nthe cost of this group is equal to n. A similar maximality result is proved\nfor generalized shift actions. Finally a study is initiated of the class of\nresidually finite, countable groups for which the finite actions are dense in\nthe space of measure preserving actions.", "doi": "10.1007/s11856-011-0182-6", "issn": "0021-2172", "publisher": "Springer", "publication": "Israel Journal of Mathematics", "publication_date": "2012-06", "series_number": "1", "volume": "189", "issue": "1", "pages": "461-507" }, { "id": "authors:nbhb2-rc397", "collection": "authors", "collection_id": "nbhb2-rc397", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130301-104948205", "type": "article", "title": "Trigonometric series and set theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This is a short historical survey concerning the interactions between the\ntheory of trigonometric series and descriptive set theory. We concentrate\nhere on the area related to problems of uniqueness for trigonometric\nseries. Detailed historical and bibliographical references can be found in the\nbooks and survey papers listed at the end.", "issn": "2080-5519", "publisher": "Polskie Towarzystwo Matematyczne", "publication": "Roczniki. Seria 2: Wiadomo\u015bci Matematyczne", "publication_date": "2012", "series_number": "2", "volume": "48", "issue": "2", "pages": "109-118" }, { "id": "authors:dqgqx-nvk43", "collection": "authors", "collection_id": "dqgqx-nvk43", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121126-103335875", "type": "article", "title": "Dynamical properties of the automorphism groups of the random poset and random distributive lattice", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Soki\u0107", "given_name": "Miodrag", "clpid": "Soki\u0107-M" } ], "abstract": "A method is developed for proving non-amenability of certain automorphism groups of countable structures and is used to show that the automorphism groups of the random poset and random distributive lattice are not amenable. The universal minimal flow of the automorphism group of the random distributive lattice is computed as a canonical space of linear orderings but it is also shown that the class of finite distributive lattices does not admit hereditary order expansions with the Amalgamation Property.", "doi": "10.4064/fm218-1-4", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "2012", "series_number": "1", "volume": "218", "issue": "1", "pages": "69-94" }, { "id": "authors:3rb7n-cbz70", "collection": "authors", "collection_id": "3rb7n-cbz70", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-103208698", "type": "article", "title": "In Memoriam: Gregory Hjorth, 1963-2011", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Greg Hjorth suddenly and unexpectedly passed away on January 13, 2011 in Melbourne, at the age of 47, due to a heart attack. He was a remarkable person, a chess prodigy who competed internationally at a high level until his early 20s, then devoted himself to the study of philosophy and mathematics and went on to become a leading figure in the field of mathematical logic and its applications.", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "2011-09", "series_number": "3", "volume": "17", "issue": "3", "pages": "471-477" }, { "id": "authors:49b17-8qt63", "collection": "authors", "collection_id": "49b17-8qt63", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090908-083705826", "type": "article", "title": "Subequivalence relations and positive-definite functions", "author": [ { "family_name": "Ioana", "given_name": "Adrian", "clpid": "Ioana-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tsankov", "given_name": "Todor", "clpid": "Tsankov-T" } ], "abstract": "We study a positive-definite function associated with a countable, measure-preserving equivalence relation, which can be used to measure quantitatively the proximity of subequivalence relations. Combined with a co-inducing construction introduced by Epstein and earlier work of Ioana, this can be used to construct many mixing actions of countable groups and establish the non-classifiability, in a strong sense, of orbit equivalence of actions of non-amenable groups. We also discuss connections with percolation on Cayley graphs and the theory of costs.", "doi": "10.48550/arXiv.0806.0430", "issn": "1661-7207", "publisher": "European Mathematical Society", "publication": "Groups, Geometry, and Dynamics", "publication_date": "2009", "series_number": "4", "volume": "3", "issue": "4", "pages": "579-625" }, { "id": "authors:88p4a-x6g86", "collection": "authors", "collection_id": "88p4a-x6g86", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-111643755", "type": "article", "title": "Means on equivalence relations", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Miller", "given_name": "Benjamin D.", "clpid": "Miller-B-D" } ], "abstract": "Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth.", "doi": "10.1007/s11856-008-0011-8", "issn": "0021-2172", "publisher": "Springer Verlag", "publication": "Israel Journal of Mathematics", "publication_date": "2008-01", "series_number": "1", "volume": "163", "issue": "1", "pages": "241-262" }, { "id": "authors:2zkan-3mb89", "collection": "authors", "collection_id": "2zkan-3mb89", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KECpams08", "type": "article", "title": "Amenable actions and almost invariant sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tsankov", "given_name": "Todor", "clpid": "Tsankov-T" } ], "abstract": "In this paper, 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 generalized Bernoulli shift, i.e., the corresponding shift action of \u0393 on M^X, where M is a measure space. In particular, we show that the action of \u0393 on X is amenable iff the shift \u0393 \u2192 M^X has almost invariant sets.", "doi": "10.1090/S0002-9939-07-09116-2", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "2008", "series_number": "2", "volume": "136", "issue": "2", "pages": "687-697" }, { "id": "authors:34s0v-by239", "collection": "authors", "collection_id": "34s0v-by239", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-084128445", "type": "article", "title": "Turbulence, amalgamation, and generic automorphisms of homogeneous structures", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Rosendal", "given_name": "Christian", "clpid": "Rosendal-C" } ], "abstract": "We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups\nof homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological\nRokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class\n(answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has\na comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups\nthat admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample\ngenerics). We show that Polish groups with ample generics have the small index property (generalizing results\nof Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable\ngroups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have\nuncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of\nmany \u03c9-stable, \u21350-categorical structures and of the random graph. In this connection, we also show that the\ninfinite symmetric group S\u221e has a unique non-trivial separable group topology. For several interesting groups\nwe also establish Serre's properties (FH) and (FA).", "doi": "10.1112/plms/pdl007", "issn": "0024-6115", "publisher": "London Mathematical Society", "publication": "Proceedings of the London Mathematical Society", "publication_date": "2007-03-01", "series_number": "2", "volume": "94", "issue": "2", "pages": "302-350" }, { "id": "authors:wwqtw-d9y58", "collection": "authors", "collection_id": "wwqtw-d9y58", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-132700025", "type": "article", "title": "Unitary representations and modular actions", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e., in a precise sense \"orthogonal\" to any Borel action of a countable group by automorphisms on a countable rooted tree. We also study tempered actions of countable groups by automorphisms on compact metrizable groups, where it turns out that this notion has several ergodic theoretic reformulations and fits naturally in a hierarchy of strong ergodicity properties strictly between ergodicity and strong mixing.", "doi": "10.1007/s10958-007-0449-y", "issn": "1072-3374", "publisher": "Springer Verlag", "publication": "Journal of Mathematical Sciences", "publication_date": "2007-01", "series_number": "3", "volume": "140", "issue": "3", "pages": "398-425" }, { "id": "authors:0xfe3-hcw27", "collection": "authors", "collection_id": "0xfe3-hcw27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130610-145818965", "type": "article", "title": "Rigidity theorems for actions of product groups and countable Borel equivalence relations", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This Memoir is both a contribution to the theory of Borel equivalence relations,\nconsidered up to Borel reducibility, and measure preserving group actions\nconsidered up to orbit equivalence. Here E is said to be Borel reducible to F if\nthere is a Borel function f with xEy if and only if f(x)Ff(y). Moreover, E is orbit\nequivalent to F if the respective measure spaces equipped with the extra structure\nprovided by the equivalence relations are almost everywhere isomorphic.\nWe consider product groups acting ergodically and by measure preserving transformations\non standard Borel probability spaces. In general terms, the basic parts\nof the monograph show that if the groups involved have a suitable notion of \"boundary\" (we make this precise with the definition of near hyperbolic), then one orbit\nequivalence relation can only be Borel reduced to another if there is some kind of\nalgebraic resemblance between the product groups and coupling of the action. This\nalso has consequence for orbit equivalence. In the case that the original equivalence\nrelations do not have non-trivial almost invariant sets, the techniques lead to relative\nergodicity results. An equivalence relation E is said to be relatively ergodic to\nF if any f with xEy\u21d2 f(x)Ff(y) has [f(x)]F constant almost everywhere.\nThis underlying collection of lemmas and structural theorems is employed in a\nnumber of different ways.\nOne of the most pressing concerns was to give completely self-contained proofs\nof results which had previously only been obtained using Zimmer's superrigidity\ntheory. We present \"elementary proofs\" that there are incomparable countable\nBorel equivalence relations (Adams-Kechris), inclusion does not imply reducibility\n(Adams), and (n + 1)E is not necessarily reducible to nE (Thomas).\nIn the later parts of the paper we give applications of the theory to specific\ncases of product groups. In particular, we catalog the actions of products of the\nfree group and obtain additional rigidity theorems and relative ergodicity results in\nthis context.\nThere is a rather long series of appendices, whose primary goal is to give the\nreader a comprehensive account of the basic techniques. But included here are also\nsome new results. For instance, we show that the Furstenberg-Zimmer lemma on\ncocycles from amenable groups fails with respect to Baire category, and use this\nto answer a question of Weiss. We also present a different proof that F_2 has the\nHaagerup approximation property.", "issn": "0065-9266", "publisher": "American Mathematical Society", "publication": "Memoirs of the American Mathematical Society", "publication_date": "2005-09", "series_number": "833", "volume": "177", "issue": "833", "pages": "1-109" }, { "id": "authors:9r886-m4x65", "collection": "authors", "collection_id": "9r886-m4x65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-135759630", "type": "article", "title": "Fra\u00efss\u00e9 Limits, Ramsey Theory, and topological dynamics of automorphism groups", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Pestov", "given_name": "V. G.", "clpid": "Pestov-P-G" }, { "family_name": "Todorcevic", "given_name": "S.", "clpid": "Todorcevic-S" } ], "abstract": "(A) In this paper we study some connections between the Fra\u00efss\u00e9 theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces X.", "doi": "10.1007/s00039-005-0503-1", "issn": "1016-443X", "publisher": "Springer", "publication": "Geometric and Functional Analysis", "publication_date": "2005-02-01", "series_number": "1", "volume": "15", "issue": "1", "pages": "106-189" }, { "id": "authors:cgh0x-enw90", "collection": "authors", "collection_id": "cgh0x-enw90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130516-152757565", "type": "article", "title": "On the classification of Polish metric spaces up to isometry", "author": [ { "family_name": "Gao", "given_name": "Su", "clpid": "Gao-Su" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study the classification problem of Polish metric spaces up to isometry and\nthe isometry groups of Polish metric spaces. In the framework of the descriptive\nset theory of definable equivalence relations, we determine the exact complexity of\nvarious classification problems concerning Polish metric spaces. We start with the\nclass of all Polish metric spaces and prove that it is Borel bireducible to the universal\norbit equivalence relation induced by Borel actions of Polish groups. We then turn\nto special classes of Polish metric spaces, including locally compact, ultrametric,\nzero-dimensional, homogeneous, and ultrahomogeneous spaces. In the investigation\nof the classification problems we also obtain characterizations for isometry groups\nof various classes of Polish metric spaces.", "issn": "0065-9266", "publisher": "American Mathematical Society", "publication": "Memoirs of the American Mathematical Society", "publication_date": "2003-01", "series_number": "766", "volume": "161", "issue": "766", "pages": "1-78" }, { "id": "authors:f9056-g4d25", "collection": "authors", "collection_id": "f9056-g4d25", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-105759569", "type": "article", "title": "Universal minimal flows of automorphism groups", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Pestov", "given_name": "V.", "clpid": "Pestov-V" }, { "family_name": "Todor\u010devi\u0107", "given_name": "S.", "clpid": "Todor\u010devi\u0107-S" } ], "abstract": "We investigate some connections between the Fra\u00efss\u00e9 theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As a result we compute the universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the \u221e-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory: the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.", "issn": "0561-7332", "publisher": "Srpska akademija nauka i umetnosti, Beograd", "publication": "Bulletin Classe des Sciences Math\u00e9matiques et Naturelles, Sciences math\u00e9matiques", "publication_date": "2003", "volume": "28", "pages": "93-106" }, { "id": "authors:6sy4f-ajc11", "collection": "authors", "collection_id": "6sy4f-ajc11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130520-104729185", "type": "article", "title": "Countable Borel equivalence relations", "author": [ { "family_name": "Jackson", "given_name": "S.", "clpid": "Jackson-S" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" } ], "abstract": "This paper develops the foundations of the descriptive set theory of countable Borel\nequivalence relations on Polish spaces with particular emphasis on the study of hyper-finite, amenable, treeable and universal equivalence relations.", "issn": "0219-0613", "publisher": "World Scientific Publishing", "publication": "Journal of Mathematical Logic", "publication_date": "2002", "series_number": "1", "volume": "2", "issue": "1", "pages": "1-80" }, { "id": "authors:egdzc-88960", "collection": "authors", "collection_id": "egdzc-88960", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111104-093324189", "type": "article", "title": "A strong generic ergodicity property of unitary and self-adjoint operators", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Sofronidis", "given_name": "N. E.", "clpid": "Sofronidis-N-E" } ], "abstract": "Consider the conjugacy action of the unitary group of an infinite-dimensional separable Hilbert space on the unitary operators. A strong generic ergodicity property of this action is established, by showing that any conjugacy invariants assigned in a definable way to unitary operators, and taking as values countable structures up to isomorphism, generically trivialize. Similar results are proved for conjugacy of self-adjoint operators and for measure equivalence. The proofs make use of the theory of turbulence for continuous actions of Polish groups, developed by Hjorth. These methods are also used to give a new solution to a problem of Mauldin in measure theory, by showing that any analytic set of pairwise orthogonal measures on the Cantor space is orthogonal to a product measure.", "doi": "10.1017/S0143385701001705", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "2001-10", "volume": "21", "pages": "1459-1479" }, { "id": "authors:3mgdb-g1w72", "collection": "authors", "collection_id": "3mgdb-g1w72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-132314741", "type": "article", "title": "Polish Metric Spaces: Their Classification and Isometry Groups", "author": [ { "family_name": "Clemens", "given_name": "John D.", "clpid": "Clemens-J-D" }, { "family_name": "Gao", "given_name": "Su", "clpid": "Gao-Su" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space (X,d). Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): \"The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not 'smooth' in the modern terminology.\" Our Theorem 2.1 below quantifies precisely the enormity of this task. After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier. The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.", "doi": "10.2307/2687754", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "2001-09", "series_number": "3", "volume": "7", "issue": "3", "pages": "361-375" }, { "id": "authors:zn712-e6r41", "collection": "authors", "collection_id": "zn712-e6r41", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-132240239", "type": "article", "title": "The prospects for mathematical logic in the twenty-first century", "author": [ { "family_name": "Buss", "given_name": "Samuel R.", "clpid": "Buss-S-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Pillay", "given_name": "Anand", "clpid": "Pillay-A" }, { "family_name": "Shore", "given_name": "Richard A.", "clpid": "Shore-R-A" } ], "abstract": "The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.", "doi": "10.2307/2687773", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "2001-06", "series_number": "2", "volume": "7", "issue": "2", "pages": "169-196" }, { "id": "authors:2q44f-edc44", "collection": "authors", "collection_id": "2q44f-edc44", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130610-142913085", "type": "article", "title": "Recent developments in the theory of Borel reducibility", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let E_0 be the Vitali equivalence relation and E_3 the product of countably\nmany copies of E_0. Two new dichotomy theorems for Borel equivalence relations are\nproved. First, for any Borel equivalence relation E that is (Borel) reducible to E_3, either\nE is reducible to E_0 or else E_3 is reducible to E. Second, if E is a Borel equivalence\nrelation induced by a Borel action of a closed subgroup of the infinite symmetric group\nthat admits an invariant metric, then either E is reducible to a countable Borel equivalence\nrelation or else E_3 is reducible to E.\nWe also survey a number of results and conjectures concerning the global structure\nof reducibility on Borel equivalence relations.", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "2001", "series_number": "1-2", "volume": "170", "issue": "1-2", "pages": "21-52" }, { "id": "authors:866an-x9143", "collection": "authors", "collection_id": "866an-x9143", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-095650096", "type": "article", "title": "Countable structures with a fixed group of automorphisms", "author": [ { "family_name": "Camerlo", "given_name": "Riccardo", "clpid": "Camerlo-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We prove that, given a countable group G, the set of countable structures (for a suitable language L)U_G whose automorphism group is isomorphic to G is a complete coanalytic set and if G \u2244 H then U_G is Borel inseparable from U_H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures for L whose automorphism group is isomorphic to \u2124_p^\u2115 ,p a prime number, is \u03a0^1_1&\u03a3^1_1-complete.", "doi": "10.1007/BF02773566", "issn": "0021-2172", "publisher": "Hebrew University Magnes Press", "publication": "Israel Journal of Mathematics", "publication_date": "2000-12", "series_number": "1", "volume": "117", "issue": "1", "pages": "105-124" }, { "id": "authors:c39a4-aa811", "collection": "authors", "collection_id": "c39a4-aa811", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111201-082818221", "type": "article", "title": "On the Classification Problem for Rank 2 Torsion-Free Abelian Groups", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study here some foundational aspects of the classification problem for torsion-free abelian groups of finite rank. These are, up to isomorphism, the subgroups of the additive groups (Q^n, +), for some n = 1, 2, 3,.... The torsion-free abelian groups of rank \u2264 n are the subgroups of (Q^n, +).", "doi": "10.1112/S0024610700001411", "issn": "0024-6107", "publisher": "London Mathematical Society", "publication": "Journal of the London Mathematical Society", "publication_date": "2000-10", "series_number": "2", "volume": "62", "issue": "2", "pages": "437-450" }, { "id": "authors:wd197-d4m26", "collection": "authors", "collection_id": "wd197-d4m26", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130524-130949007", "type": "article", "title": "Linear algebraic groups and countable Borel equivalence relations", "author": [ { "family_name": "Adams", "given_name": "Scot", "clpid": "Adams-Scot" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper is a contribution to the study of Borel equivalence relations on standard\nBorel spaces (i.e., Polish spaces equipped with their Borel structure). In\nmathematics one often deals with problems of classification of objects up to some\nnotion of equivalence by invariants. Frequently these objects can be viewed as elements\nof a standard Borel space X and the equivalence turns out to be a Borel\nequivalence relation E on X. A complete classification of X up to E consists of\nfinding a set of invariants I and a map c : X \u2192 I such that xEy \u21d4 c(x) = c(y).\nFor this to be of any interest both I and c must be explicit or definable and as simple\nand concrete as possible. The theory of Borel equivalence relations studies the\nset-theoretic nature of possible invariants and develops a mathematical framework\nfor measuring the complexity of such classification problems.", "doi": "10.1090/S0894-0347-00-00341-6", "issn": "0894-0347", "publisher": "American Mathematical Society", "publication": "Journal of the American Mathematical Society", "publication_date": "2000-06-23", "series_number": "4", "volume": "13", "issue": "4", "pages": "909-943" }, { "id": "authors:absez-hgb80", "collection": "authors", "collection_id": "absez-hgb80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-075724313", "type": "article", "title": "The complexity of the classification of Riemann surfaces and complex manifolds", "author": [ { "family_name": "Hjorth", "given_name": "G.", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of \u2102 can be used as complete invariants for Riemann surfaces considered up to conformal equivalence, and that this equivalence relation is itself Borel in a natural Borel structure on the space of all such surfaces. We further proceed to precisely calculate the classification difficulty of this equivalence relation in terms of the modern theory of Borel equivalence relations. \n\nOn the other hand we show that the analog of Becker, Rubel, and Henson's question has a negative solution in (complex) dimension n \u2265 2.", "issn": "0019-2082", "publisher": "University of Illinois", "publication": "Illinois Journal of Mathematics", "publication_date": "2000-03", "series_number": "1", "volume": "44", "issue": "1", "pages": "104-137" }, { "id": "authors:yw7zg-jrn74", "collection": "authors", "collection_id": "yw7zg-jrn74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-095020744", "type": "article", "title": "New Directions in Descriptive Set Theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical\nexamples are R^n, C^n, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2^N, the Baire space N^N, the infinite symmetric group S_\u221e, the unitary group (of the Hilbert space), the group of\nmeasure preserving transformations of the unit interval, etc.", "doi": "10.2307/421088", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "1999-06", "series_number": "2", "volume": "5", "issue": "2", "pages": "161-174" }, { "id": "authors:mbe97-y1b24", "collection": "authors", "collection_id": "mbe97-y1b24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-100529842", "type": "article", "title": "Borel Chromatic Numbers", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Solecki", "given_name": "S.", "clpid": "Solecki-S" }, { "family_name": "Todorcevic", "given_name": "S.", "clpid": "Todorcevic-S" } ], "abstract": "We study in this paper graph coloring problems in the context of descriptive set theory. We consider graphs G=(X, R), where the vertex set X is a standard Borel space (i.e., a complete separable metrizable space equipped with its \u03c3-algebra of Borel sets), and the edge relation R \u2286 X^2 is\n\"definable\", i.e., Borel, analytic, co-analytic, etc.\n\nA Borel n-coloring of such a graph, where 1\u2a7d n \u2a7d N_0 , is a Borel map c: X \u2192 Y with card(Y)=n, such\nthat xRy\u21d2c(x) \u2260 {c( y). If such a Borel coloring exists we define the Borel\nchromatic number of G, in symbols X_B(G), to be the smallest such n.\nOtherwise we say that G has uncountable Borel chromatic number, in symbols X_B(G) > N_0.", "doi": "10.1006/aima.1998.1771", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "1999-01-15", "series_number": "1", "volume": "141", "issue": "1", "pages": "1-44" }, { "id": "authors:jzzgb-w4761", "collection": "authors", "collection_id": "jzzgb-w4761", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-113148990", "type": "article", "title": "Rigidity properties of Borel ideals on the integers", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We classify the Borel ideals I on the set of natural numbers N for which p(N)/I can be Borel embedded into the orbit space of a Borel action of the infinite symmetric group. As a consequence we show that certain Borel ideals I, including the Fr\u00e9chet ideal, are completely characterized by the \"Borel cardinality\" of the set p(N)/I.", "doi": "10.1016/S0166-8641(97)00150-8", "issn": "0166-8641", "publisher": "Elsevier", "publication": "Topology and Its Applications", "publication_date": "1998-05-22", "series_number": "1-3", "volume": "85", "issue": "1-3", "pages": "195-205" }, { "id": "authors:pryh3-whk19", "collection": "authors", "collection_id": "pryh3-whk19", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-142221264", "type": "article", "title": "Borel equivalence relations induced by actions of the symmetric group", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially \u03a0^0_3 iff the invariants are countable sets of reals, it is potentially \u03a0^0_4 iff the invariants are countable sets of countable sets of reals, and so on. The proofs use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models.", "doi": "10.1016/S0168-0072(97)00049-3", "issn": "0168-0072", "publisher": "Elsevier", "publication": "Annals of Pure and Applied Logic", "publication_date": "1998-03-11", "series_number": "1", "volume": "92", "issue": "1", "pages": "63-112" }, { "id": "authors:tjnvz-v5872", "collection": "authors", "collection_id": "tjnvz-v5872", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-105907286", "type": "article", "title": "Measurable enumeration of eigenelements", "author": [ { "family_name": "Gordon", "given_name": "Alexander Y.", "clpid": "Gordon-A-Y" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We prove that for eigenelements of a measurable family of linear self-adjoint operators in a separable Hilbert space there exists a measurable enumeration. We also prove a similar result for measurable families of bounded linear operators having at most countably many eigenvalues (under certain restrictions on the parameter space). The proof of the latter result is based on descriptive set theory, while in the case of self-adjoint (and some more general) operators the proof is constructive.", "issn": "0003-6811", "publisher": "Taylor and Francis", "publication": "Applicable Analysis: An International Journal", "publication_date": "1998", "series_number": "1-4", "volume": "71", "issue": "1-4", "pages": "41-61" }, { "id": "authors:2c6jq-p1053", "collection": "authors", "collection_id": "2c6jq-p1053", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130520-144530065", "type": "article", "title": "New Dichotomies for Borel Equivalence Relations", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.", "doi": "10.2307/421148", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "1997-09", "series_number": "3", "volume": "3", "issue": "3", "pages": "329-346" }, { "id": "authors:fzwwc-ewv74", "collection": "authors", "collection_id": "fzwwc-ewv74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-135837396", "type": "article", "title": "On the concept of \u03c0^1_1-completeness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is shown that two natural notions of completeness for co-analytic sets in Polish spaces, one in terms of continuous reductions and the other in terms of Borel reductions, coincide. The proof uses methods of effective descriptive set theory.", "doi": "10.1090/S0002-9939-97-03770-2", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1997-06", "series_number": "6", "volume": "125", "issue": "6", "pages": "1811-1814" }, { "id": "authors:56k1v-f9h12", "collection": "authors", "collection_id": "56k1v-f9h12", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-133826211", "type": "article", "title": "The classification of hypersmooth Borel equivalence relations", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel auto-morphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bi-reducibility, and Borel isomorphism.", "doi": "10.1090/S0894-0347-97-00221-X", "issn": "0894-0347", "publisher": "American Mathematical Society", "publication": "Journal of the American Mathematical Society", "publication_date": "1997-01", "series_number": "1", "volume": "10", "issue": "1", "pages": "215-242" }, { "id": "authors:mnx01-0pa63", "collection": "authors", "collection_id": "mnx01-0pa63", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-141337004", "type": "article", "title": "Borel equivalence relations and classifications of countable models", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Using the theory of Borel equivalence relations we analyze the isomorphism relation on the countable models of a theory and develop a framework for measuring the complexity of possible complete invariants for isomorphism.", "doi": "10.1016/S0168-0072(96)00006-1", "issn": "0168-0072", "publisher": "Elsevier", "publication": "Annals of Pure and Applied Logic", "publication_date": "1996-12-15", "series_number": "3", "volume": "82", "issue": "3", "pages": "221-272" }, { "id": "authors:krf6r-wtm04", "collection": "authors", "collection_id": "krf6r-wtm04", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130520-160211996", "type": "article", "title": "Analytic Equivalence Relations and Ulm-Type Classifications", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.", "doi": "10.2307/2275888", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1995-12", "series_number": "4", "volume": "60", "issue": "4", "pages": "1273-1300" }, { "id": "authors:bjnyy-epy71", "collection": "authors", "collection_id": "bjnyy-epy71", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-103730289", "type": "article", "title": "Approximation of analytic by Borel sets and definable countable chain conditions", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Solecki", "given_name": "S.", "clpid": "Solecki-S" } ], "abstract": "Let I be a \u03c3-ideal on a Polish space such that each set from I is contained in a Borel set from I. We say that I fails to fulfil the \u03a3_1^1 countable chain condition if there is a \u03a3_1^1 equivalence relation with uncountably many equivalence classes none of which is in I. Assuming definable determinacy, we show that if the family of Borel sets from I is definable in the codes of Borel sets, then each \u03a3_1^1 set is equal to a Borel set modulo a set from I iff I fulfils the \u03a3_1^1 countable chain condition. Further we characterize the \u03c3-ideals I generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property for \u03a3_1^1 sets mentioned above. It turns out that they are exactly of the form MGR(F)={A : \u2200F \u2208 F A \u2229F is meager in F} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the \u03c3-ideal of meager sets is the unique \u03c3-ideal on R, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition.", "doi": "10.1007/BF02808208", "issn": "0021-2172", "publisher": "Hebrew University Magnes Press", "publication": "Israel Journal of Mathematics", "publication_date": "1995-10", "series_number": "1-3", "volume": "89", "issue": "1-3", "pages": "343-356" }, { "id": "authors:p3p13-xp104", "collection": "authors", "collection_id": "p3p13-xp104", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-110331634", "type": "article", "title": "Topology and descriptive set theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. \nInstead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Borel actions of Polish groups and Borel equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory).\nThere are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Borel actions. The last part concentrates on Borel equivalence relations.\nThe exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline.", "doi": "10.1016/0166-8641(94)00146-4", "issn": "0166-8641", "publisher": "Elsevier", "publication": "Topology and Its Applications", "publication_date": "1994-08-15", "series_number": "3", "volume": "58", "issue": "3", "pages": "195-222" }, { "id": "authors:2whnz-a9y96", "collection": "authors", "collection_id": "2whnz-a9y96", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130524-135334590", "type": "article", "title": "Countable Sections for Locally Compact Group Actions. II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "In this paper we study the structure of the orbit equivalence relation induced by a Borel action of a second countable locally compact group on a standard Borel space.", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1994-01", "series_number": "1", "volume": "120", "issue": "1", "pages": "241-247" }, { "id": "authors:r3tcq-5nk56", "collection": "authors", "collection_id": "r3tcq-5nk56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-092754518", "type": "article", "title": "The structure of hyperfinite Borel equivalence relations", "author": [ { "family_name": "Dougherty", "given_name": "R.", "clpid": "Dougherty-R" }, { "family_name": "Jackson", "given_name": "S.", "clpid": "Jackson-S" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.", "doi": "10.2307/2154620", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1994-01", "series_number": "1", "volume": "341", "issue": "1", "pages": "193-225" }, { "id": "authors:hgkgt-0dm64", "collection": "authors", "collection_id": "hgkgt-0dm64", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-103113640", "type": "article", "title": "Amenable Versus Hyperfinite Borel Equivalence Relations", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let X be a standard Borel space (i.e., a Polish space with the associated Borel structure), and let E be a countable Borel equivalence relation on X, i.e., a Borel equivalence relation E for which every equivalence class [X]_E is countable. By a result of Feldman-Moore [FM], E is induced by the orbits of a Borel action of a countable group G on X.", "doi": "10.2307/2275102", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1993-09", "series_number": "3", "volume": "58", "issue": "3", "pages": "894-907" }, { "id": "authors:raqc2-nra14", "collection": "authors", "collection_id": "raqc2-nra14", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130523-095229531", "type": "article", "title": "Borel actions of Polish groups", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures and the topological Vaught conjecture. We establish the existence of universal actions for any given Polish group, extending a result of Mackey and Varadarajan for the locally compact case. We prove an analogue of Tarski's theorem on paradoxical decompositions, by showing that the existence of an invariant Borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many Borel pieces. We show that various natural versions of the topological Vaught conjecture are equivalent to each other and, in the case of the group of permutations of N, with the model-theoretic Vaught conjecture for infinitary logic; this depends on our identification of the universal action for that group.", "doi": "10.1090/S0273-0979-1993-00383-5", "issn": "0273-0979", "publisher": "American Mathematical Society", "publication": "Bulletin of the American Mathematical Society", "publication_date": "1993-04", "series_number": "2", "volume": "28", "issue": "2", "pages": "334-341" }, { "id": "authors:1b42s-01575", "collection": "authors", "collection_id": "1b42s-01575", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130516-154918991", "type": "article", "title": "Countable sections for locally compact group actions", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It has been shown by J. Feldman, P. Hahn and C. C. Moore that every\nnon-singular action of a second countable locally compact group has a countable\n(in fact so-called lacunary) complete measurable section. This is extended here to\nthe purely Borel theoretic category, consisting of a Borel action of such a group on\nan analytic Borel space (without any measure). Characterizations of when an\narbitrary Borel equivalence relation admits a countable complete Borel section are\nalso established.", "doi": "10.1017/S0143385700006751", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "1992-06-01", "series_number": "2", "volume": "12", "issue": "2", "pages": "283-295" }, { "id": "authors:0kbz6-x3q75", "collection": "authors", "collection_id": "0kbz6-x3q75", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-082635644", "type": "article", "title": "Descriptive Set Theory and Harmonic Analysis", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" } ], "abstract": "During the 1989 European ASL Summer Meeting in Berlin, the\nauthors gave a series of eight lectures (short course) on the topic of the title. This\nsurvey article consists basically of the lecture notes for that course distributed to\nthe participants of that conference. We have purposely tried in this printed version\nto preserve the informal style of the original notes.", "doi": "10.2307/2275277", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1992-06", "series_number": "2", "volume": "57", "issue": "2", "pages": "413-441" }, { "id": "authors:bewz5-6x728", "collection": "authors", "collection_id": "bewz5-6x728", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-102333066", "type": "article", "title": "The Complexity of Antidifferentiation", "author": [ { "family_name": "Dougherty", "given_name": "Randall", "clpid": "Dougherty-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We consider real-valued functions defined on the interval [0, 1]. We denote by \u0394 the set of derivatives; i.e., \u0192 \u0404 \u0394 iff there is a differentiable function F such that F' = \u0192. Any such F is a primitive of \u0192 and is uniquely determined up to a constant. To normalize, we denote by F(x)=\u0192^x_0\u0192 the\nprimitive determined by F(0)=0. This is the original Newtonian concept of integration as antidifferentiation.", "doi": "10.1016/0001-8708(91)90006-S", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "1991-08", "series_number": "2", "volume": "88", "issue": "2", "pages": "145-169" }, { "id": "authors:q6sdf-63e75", "collection": "authors", "collection_id": "q6sdf-63e75", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-093201219", "type": "article", "title": "Hereditary properties of the class of closed sets of uniqueness for trigonometric series", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is shown that the\u03c3-ideal U_0 of closed sets of extended uniqueness in T is hereditarily non-Borel, i.e. every \"non-trivial\" \u03c3-ideal of closed sets I \u2286 U_0 is non-Borel. This implies both the result of Solovay, Kaufman that both U_0 and U(the \u03c3-ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset of T of extended uniqueness is of the first category. A further extension to ideals contained in U_0 is given.", "doi": "10.1007/BF02772948", "issn": "0021-2172", "publisher": "Hebrew University Magnes Press", "publication": "Israel Journal of Mathematics", "publication_date": "1991-06", "series_number": "2", "volume": "73", "issue": "2", "pages": "189-198" }, { "id": "authors:sfhk2-sa892", "collection": "authors", "collection_id": "sfhk2-sa892", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-103738099", "type": "article", "title": "Boundedness theorems for dilators and ptykes", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The main theorem of this paper is: If \u0192 is a partial function from \u2135_1 to \u2135_1 which is \u2211^1_1-bounded, then there is a weakly finite primitive recursive dilator D such that for all infinite \u03b1 \u03f5 dom(\u0192), \u0192(\u03b1) \u2a7d D(\u03b1). The proof involves only elementary combinatorial constructions of trees. A generalization to ptykes is also given.", "doi": "10.1016/0168-0072(91)90040-S", "issn": "0168-0072", "publisher": "Elsevier Masson", "publication": "Annals of Pure and Applied Logic", "publication_date": "1991-04-15", "series_number": "1-2", "volume": "52", "issue": "1-2", "pages": "79-92" }, { "id": "authors:3b1ft-4qg60", "collection": "authors", "collection_id": "3b1ft-4qg60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-105346791", "type": "article", "title": "A strong boundedness theorem for dilators", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Woodin", "given_name": "W. H.", "clpid": "Woodin-W-H" } ], "abstract": "We prove a strong boundedness theorem for dilators: if A \u2286 DIL is \u03a3^1_1, then there is a recursive dilator D_0 such that \u2200D \u2208 A (D can be embedded into D_0).", "doi": "10.1016/0168-0072(91)90041-J", "issn": "0168-0072", "publisher": "Elsevier Masson", "publication": "Annals of Pure and Applied Logic", "publication_date": "1991-04-15", "series_number": "1-2", "volume": "52", "issue": "1-2", "pages": "93-97" }, { "id": "authors:q6c4c-5ar39", "collection": "authors", "collection_id": "q6c4c-5ar39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-150404346", "type": "article", "title": "Amenable Equivalence Relations and Turing Degrees", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "In [12] Slaman and Steel posed the following problem:\nAssume ZF + DC + AD. Suppose we have a function assigning to each Turing\ndegree d a linear order <_d of d. Then must the rationals embed order preservingly in\n<_d for a cone of d's?", "doi": "10.2307/2274913", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1991-03", "series_number": "1", "volume": "56", "issue": "1", "pages": "182-194" }, { "id": "authors:rc9pp-sqd71", "collection": "authors", "collection_id": "rc9pp-sqd71", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-150609710", "type": "article", "title": "The Class Of Synthesizable Pseudomeasures", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" }, { "family_name": "Tardivel", "given_name": "V.", "clpid": "Tardivel-V" } ], "abstract": "In this paper we study descriptive set theoretic questions related to\nconcepts of harmonic synthesis on the unit circle T, and their relationship\nwith the structure of uniqueness sets.", "issn": "0019-2082", "publisher": "Illinois Journal of Mathematics", "publication": "Illinois Journal of Mathematics", "publication_date": "1991-03", "series_number": "1", "volume": "35", "issue": "1", "pages": "107-146" }, { "id": "authors:cvb33-qt212", "collection": "authors", "collection_id": "cvb33-qt212", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-131913642", "type": "article", "title": "A Glimm-Effros dichotomy for Borel equivalence relations", "author": [ { "family_name": "Harrington", "given_name": "L. A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" } ], "abstract": "A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in the Polish group action\ncase when additionally the induced equivalence relation is F\u03c3. It is the purpose of this paper to extend the Glimm-Effros dichotomy to the very general context of an arbitrary Borel equivalence relation (not even necessarily induced by a group action). Despite the totally classical descriptive set-theoretic nature of our result, our proof requires the employment of methods of effective descriptive\nset theory and thus ultimately makes crucial use of computability (or recursion) theory on the integers.", "doi": "10.1090/S0894-0347-1990-1057041-5", "issn": "0894-0347", "publisher": "American Mathematical Society", "publication": "Journal of the American Mathematical Society", "publication_date": "1990-10", "series_number": "4", "volume": "3", "issue": "4", "pages": "903-928" }, { "id": "authors:xtm8s-6zw43", "collection": "authors", "collection_id": "xtm8s-6zw43", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-151355415", "type": "article", "title": "A Classification of Baire Class 1 Functions", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" } ], "abstract": "We study in this paper various ordinal ranks of (bounded) Baire class 1 functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class 1 functions B_1 in a transfinite hierarchy B^\u03be_1 \u03be < \u03c9_1) of \"small\" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of \u0394^0_2 sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class 1 function \u0192 are introduced and used in this study.", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1990-03", "series_number": "1", "volume": "318", "issue": "1", "pages": "209-236" }, { "id": "authors:ryk0a-81x05", "collection": "authors", "collection_id": "ryk0a-81x05", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-084032401", "type": "article", "title": "Covering theorems for uniqueness and extended uniqueness sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "A covering theorem for a class of sets C asserts that every set in C can be covered by a countable union of sets in some (somehow simpler) class C. In the theory of sets of uniqueness on the unit circle T the first result of this\nkind is Piatetski-Shapiro's theorem in [PS], which states that every closed set of uniqueness can be covered by countably many closed sets in the class U'_1, consisting of those closed sets E \u2286 T for which there exists a sequence of\nfunctions in A(T), vanishing on E, which converges to the function 1 in the weak^*-topology.", "issn": "0010-1354", "publisher": "Polish Academy of Sciences, Institute of Mathematics", "publication": "Colloquium Mathematicum", "publication_date": "1990", "series_number": "1", "volume": "59", "issue": "1", "pages": "63-79" }, { "id": "authors:x6qt8-jqp29", "collection": "authors", "collection_id": "x6qt8-jqp29", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-084327264", "type": "article", "title": "\u03a0^1_1 Borel Sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Marker", "given_name": "David", "clpid": "Marker-D" }, { "family_name": "Sami", "given_name": "Ramez L.", "clpid": "Sami-R-L" } ], "abstract": "The results in this paper were motivated by the following\nquestion of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal \u0251 such that all models of T have Scott rank below \u0251? If Martin's conjecture is true for T then \u0251 \u2264 \u03c9\u00b72.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1989-09", "series_number": "3", "volume": "54", "issue": "3", "pages": "915-920" }, { "id": "authors:wwr0g-prk26", "collection": "authors", "collection_id": "wwr0g-prk26", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130524-113257744", "type": "article", "title": "Hausdorff Measures and Sets of Uniqueness for Trigonometric Series", "author": [ { "family_name": "Dougherty", "given_name": "R.", "clpid": "Dougherty-R" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We characterize the closed sets E in the unit circle T which have the property that, for some nondecreasing h: (0, \u221e) \u2192(0, \u221e) with h(0+) = 0, all the Hausdorff h-measure 0 closed sets F \u2286 E are sets of uniqueness (for trigonometric series). In conjunction with K\u00f6rner's result on the existence of Helson sets of multiplicity, this implies the existence of closed sets of multiplicity (M-sets) within which Hausdorff h-measure 0 implies uniqueness, for some h. This is contrasted with the case of closed sets of strict multiplicity (M_0-sets), where results of Ivashev-Musatov and Kaufman establish the opposite.", "doi": "10.2307/2047049", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1989-04", "series_number": "4", "volume": "105", "issue": "4", "pages": "894-897" }, { "id": "authors:wj2dt-qcf58", "collection": "authors", "collection_id": "wj2dt-qcf58", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-131009335", "type": "article", "title": "Ordinal Rankings on Measures Annihilating Thin Sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Lyons", "given_name": "Russell", "clpid": "Lyons-R" } ], "abstract": "We assign a countable ordinal number to each probability measure which annihilates all H-sets. The descriptive-set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non-Borel. In addition, it allows us to quantify the failure of Rajchman's conjecture. Similar results are obtained for measures annihilating Dirichlet sets.", "doi": "10.2307/2000989", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1988-12", "series_number": "2", "volume": "310", "issue": "2", "pages": "747-758" }, { "id": "authors:hpg12-cvn82", "collection": "authors", "collection_id": "hpg12-cvn82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-115611198", "type": "article", "title": "The Set of Continuous Functions with the Everywhere Convergent Fourier Series", "author": [ { "family_name": "Ajtai", "given_name": "M.", "clpid": "Ajtai-M" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper deals with the descriptive set theoretic properties of the class EC of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in C(T). A natural coanalytic rank function on EC is studied that assigns to each \u0192 \u0404 EC a countable ordinal number, which measures the \"complexity\" of the convergence of the Fourier series of \u0192. It is shown that there exist functions in EC (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on EC with \u03c9_1 distinct levels.", "doi": "10.2307/2000906", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1987-07", "series_number": "1", "volume": "302", "issue": "1", "pages": "207-221" }, { "id": "authors:ymnan-6h009", "collection": "authors", "collection_id": "ymnan-6h009", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-103643006", "type": "article", "title": "The Structure of \u03c3-Ideals of Compact Sets", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" }, { "family_name": "Woodin", "given_name": "W. H.", "clpid": "Woodin-W-H" } ], "abstract": "Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where \u03c3-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of \u03c3-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a \u03c3-ideal imposes severe definability\nrestrictions. A typical instance is the dichotomy theorem, which states that \u03c3-ideals which are analytic or coanalytic must be actually either complete coanalytic or else G_\u03b4. In the second part we discuss (generators or as we call\nthem here) bases for \u03c3-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel \u03c3-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the\nthird part we develop the connections of the definability properties of \u03c3-ideals with other structural properties, like the countable chain condition, etc.", "doi": "10.2307/2000338", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1987-05", "series_number": "1", "volume": "301", "issue": "1", "pages": "263-288" }, { "id": "authors:xaers-p8324", "collection": "authors", "collection_id": "xaers-p8324", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-145231519", "type": "article", "title": "Ranks of differentiable functions", "author": [ { "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" } ], "abstract": "The purpose of this paper is to define and study a natural rank function\nwhich associates to each differentiable function (say on the interval [0, 1]) a\ncountable ordinal number, which measures the complexity of its derivative.\nFunctions with continuous derivatives have the smallest possible rank 1, a\nfunction like x^2 sin (x^(-1)) has rank 2, etc., and we show that functions of any\ngiven countable ordinal rank exist. This exhibits an underlying hierarchical\nstructure of the class of differentiable functions, consisting of \u03c9_1 distinct levels.\nThe definition of rank is invariant under addition of constants, and so it\nnaturally assigns also to every derivative a unique rank, and an associated\nhierarchy for the class of all derivatives.\n\nThe set D of functions in C[0, 1] which are everywhere differentiable is a\ncomplete coanalytic (and thus non-Borel) set (Mazurkiewicz [Maz]; see Section\n2 below) and it will tum out that the rank function we define has the\nright descriptive set theoretic properties summarized in the concept of a\ncoanalytic norm, explained in Section 1.\n\nOur original description of the rank function was in terms of wellfounded\ntrees and is given in Section 4. In Section 3 we give an equivalent description\nin terms of a Cantor-Bendixson type analysis. We would like to acknowledge\nhere the contribution of D. Preiss. It was in a conversation with one of the\nauthors that this equivalent description was formulated.", "doi": "10.1112/S0025579300011244", "issn": "0025-5793", "publisher": "University College London", "publication": "Mathematika", "publication_date": "1986-12", "series_number": "2", "volume": "33", "issue": "2", "pages": "252-278" }, { "id": "authors:k35mp-zkr91", "collection": "authors", "collection_id": "k35mp-zkr91", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-101108599", "type": "article", "title": "On the Relative Consistency Strength of Determinacy Hypothesis", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Solovay", "given_name": "Robert M.", "clpid": "Solovay-R-M" } ], "abstract": "For any collection of sets of reals C, let C-DET be the statement that all sets of reals in C are determined. In this paper we study questions of the form: For given C \u2286 C', when is C'-DET equivalent, equiconsistent or strictly stronger in consistency strength than C-DET (modulo ZFC)? We focus especially on classes C contained in the projective sets.", "doi": "10.2307/1999790", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1985-07", "series_number": "1", "volume": "290", "issue": "1", "pages": "179-211" }, { "id": "authors:zx0q4-q1d11", "collection": "authors", "collection_id": "zx0q4-q1d11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-152728059", "type": "article", "title": "Determinacy with Complicated Strategies", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "For any class of functions F from R into R, AD(F) is the assertion that in every two person game on integers one of the two players has a winning strategy in the class F. It is shown, in ZF + DC + V = L(R), that for any F of cardinality < 2^(N0)(i.e. any F which is a surjective image of R) AD(F) implies AD (the Axiom of Determinacy).", "doi": "10.1090/S0002-9939-1985-0784188-0", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1985-06", "series_number": "2", "volume": "94", "issue": "2", "pages": "333-336" }, { "id": "authors:afzqs-0pg39", "collection": "authors", "collection_id": "afzqs-0pg39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-083650112", "type": "article", "title": "The Axiom of Determinacy Implies Dependent Choices in L(R)", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We prove the following Main Theorem: ZF+AD+V=L(R)\u21d2DC. As a corollary we have that Con(ZF+AD)\u21d2Con(ZF+AD+DC). Combined with the result of Woodin that Con(ZF+AD)\u21d2Con(ZF+AD+\u00acAC^\u03c9) it follows that DC (as well as AC^\u03c9) is independent relative to ZF+AD. It is finally shown (jointly with H. Woodin) that ZF+AD+\u00acDC_R, where DC_R is DC restricted to reals, implies the consistency of ZF+AD+DC, in fact implies R^# (i.e. the sharp of L(R)) exists.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1984-03", "series_number": "1", "volume": "49", "issue": "1", "pages": "161-173" }, { "id": "authors:6x4k8-ghn42", "collection": "authors", "collection_id": "6x4k8-ghn42", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KECpnas83", "type": "article", "title": "Equivalence of partition properties and determinacy", "author": [ { "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" } ], "abstract": "It is shown that, within L(R), the smallest inner model of set theory containing the reals, the axiom of determinacy is equivalent to the existence of arbitrarily large cardinals below \u0398 with the strong partition property \u0138 \u2192 (\u0138)^\u0138.", "pmcid": "PMC393690", "issn": "0027-8424", "publisher": "National Academy of Sciences", "publication": "Proceedings of the National Academy of Sciences of the United States of America", "publication_date": "1983-03-15", "series_number": "6", "volume": "80", "issue": "6", "pages": "1783-1786" }, { "id": "authors:1ttt9-hw631", "collection": "authors", "collection_id": "1ttt9-hw631", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-105719531", "type": "article", "title": "Forcing with \u0394 perfect trees and minimal \u0394-degrees", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper is a sequel to [3] and it contains, among other things, proofs of the results announced in the last section of that paper. In \u00a71, we use the general method of [3] together with reflection arguments to study the properties of forcing with \u0394 perfect trees, for certain Spector pointclasses \u0393, obtaining as a main result the existence of a continuum of minimal \u0394-degrees for such \u0393's, under determinacy hypotheses. In particular, using PD, we prove the existence of continuum many minimal \u0394^(1)_(2n+1)-degrees, for all n.^(2) Following an idea of Leo Harrington, we extend these results in \u00a72 to show the existence of minimal strict upper bounds for sequences of \u0394-degrees which are not too far apart. As a corollary, it is computed that the length of the natural hierarchy of \u0394^(1)_(2n+1)-degrees is equal to \u03c9 when n \u2265 1. (By results of Sacks and Richter the length of the natural hierarchy of \u0394^(1)_(1)-degrees is known to be equal to the first recursively inaccessible ordinal.)", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1981-12", "series_number": "4", "volume": "46", "issue": "4", "pages": "803-816" }, { "id": "authors:m56ts-pbx63", "collection": "authors", "collection_id": "m56ts-pbx63", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-085823503", "type": "article", "title": "On the determinacy of games on ordinals", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let \u03bb be an ordinal and A \u2286 \u03bb^\u03c9 x \u03bb^\u03c9. As usual we associate with it the game G(A;\u03bb): I II I and II alternatively \u03be0 \u03b70 play \u03be0, \u03b70, \u03be1,\u03b71,.... from \u03bb; \u03be1 I wins iff (\u03be, \u03ae) \u03b5A.\n\u03b71 \u03be \u03ae.", "doi": "10.1016/0003-4843(81)90001-2", "issn": "0003-4843", "publisher": "Elsevier", "publication": "Annals of Mathematical Logic", "publication_date": "1981-06", "series_number": "2", "volume": "20", "issue": "2", "pages": "109-154" }, { "id": "authors:0ypgb-1hd98", "collection": "authors", "collection_id": "0ypgb-1hd98", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130610-140456846", "type": "article", "title": "An overview of descriptive set theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Understanding the structure of sets of reals is one of the\nfundamental problems of set theory.\nOf course, an arbitrary subset of the continuum is immensely complicated,\nand exhibits in general no regularities. For example it could be non(Lebesgue)-measurable\nor fail to have the property of Baire. So from the early days of\nset theory, about a hundred years ago, mathematicians have concentrated on the\nstudy of various classes of definable sets of reals, wide enough to include the\nusual sets appearing in analysis, topology, etc., but also restricted to possibly\nexhibit a regular behaviour.\nThe study of such classes of definable sets is the purpose of descriptive\nset theory. Indeed, to a large extent descriptive set theory is mainly concerned\nwith the structure of the projective sets of reals, which we proceed to define.", "issn": "1151-1745", "publisher": "Universit\u00e9 Pierre et Marie Curie", "publication": "Publications Math\u00e9matiques de l'Universit\u00e9 Pierre et Marie Curie", "publication_date": "1979-02", "volume": "4" }, { "id": "authors:4c62c-cpc60", "collection": "authors", "collection_id": "4c62c-cpc60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-081912845", "type": "article", "title": "The Perfect Set Theorem and Definable Wellorderings of the Continuum", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let \u0393 be a collection of relations on the reals and let M be a set of reals. We call M a perfect set basis for \u0393 if every set in \u0393 with parameters from M which is not totally included in M contains a perfect subset with code in M. A simple elementary proof is given of the following result (assuming mild regularity conditions on \u0393 and M): If M is a perfect set basis for \u0393, the field of every wellordering in \u0393 is contained in M. An immediate corollary is Mansfield's Theorem that the existence of a \u03a3^1_2 wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1978-12", "series_number": "4", "volume": "43", "issue": "4", "pages": "630-634" }, { "id": "authors:9ewfk-rwc14", "collection": "authors", "collection_id": "9ewfk-rwc14", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-081838953", "type": "article", "title": "Countable ordinals and the analytical hierarchy, II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper is a contribution to the study of projective sets under the hypothesis of definable determinacy. For our purposes here this can be understood as the hypothesis every set of rea1s in L[\u03c9^\u03c9] (= the smallest inner model of ZF containing \u03c9^\u03c9) is determined. Since questions about the projective hierarchy are absolute between the real world and\nL[\u03c9^\u03c9] it is innocuous to work entirely within L[\u03c9^\u03c9].", "doi": "10.1016/0003-4843(78)90010-4", "issn": "0003-4843", "publisher": "Elsevier", "publication": "Annals of Mathematical Logic", "publication_date": "1978-12", "series_number": "3", "volume": "15", "issue": "3", "pages": "193-223" }, { "id": "authors:rbpcw-3vz95", "collection": "authors", "collection_id": "rbpcw-3vz95", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-083000082", "type": "article", "title": "Minimal Upper Bounds for Sequences of \u0394^1_(2n)-Degrees", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is proved here, assuming Projective Determinacy, that every ascending sequence of \u0394^1_(2n)-degrees has a minimal strict upper bound but no least strict upper bound. This generalizes a result of Friedman for n = 1.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1978-09", "series_number": "3", "volume": "43", "issue": "3", "pages": "502-507" }, { "id": "authors:x826v-knz27", "collection": "authors", "collection_id": "x826v-knz27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130625-140857081", "type": "article", "title": "On the theory of \u220f_3^1 sets of reals", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "D. A.", "clpid": "Martin-D-A" } ], "abstract": "Assuming that \u2200x \u0404 \u03c9^\u03c9 (x^# exists), let u_\u0251 be the \u0251th uniform indiscernible (see [3] or [2] ). A canonical coding system for ordinals < u_\u03c9 can be defined by letting W0_\u03c9 = {w \u0404 \u03c9^\u03c9: w = (n, x^#), for some n \u0404 \u03c9, x \u0404 \u03c9^\u03c9} and for w = (n, x^#) \u0454 W0_\u03c9, \u2502w\u2502 = \u01ac^L_n [x](u_l',... , u_k_n),\nwhere T_n is the nth term in a recursive enumeration of all terms in the language of ZF + V = L [x], x a constant, taking always ordinal values. Call a relation P(\u03be x), where ~varies over u^\u03c9 and x over \u03c9^\u03c9, \u220f^1_k if P^*(w, x)\u21d4 w \u0404 W0_\u03c9 \u039b P(\u2502w\u2502, x) is \u220f^1_k. An ordinal \u03be < u_\u03c9 is called \u0394^1_k if it has a \u0394^1_k notation i.e. \u2203 w \u0404 W0_\u03c9 (w \u0404 \u0394^1_k \u039b \u2502w\u2502 = \u03be).", "doi": "10.1090/S0002-9904-1978-14447-4", "issn": "0273-0979", "publisher": "American Mathematical Society", "publication": "Bulletin of the American Mathematical Society", "publication_date": "1978-01", "series_number": "1", "volume": "84", "issue": "1", "pages": "149-151" }, { "id": "authors:vymew-27z56", "collection": "authors", "collection_id": "vymew-27z56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-135105483", "type": "article", "title": "On a notion of smallness for subsets of the Baire space", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let us call a set A \u2286 \u03c9^\u03c9 of functions from \u03c9 into \u03c9 \u03c3-bounded if there is a countable sequence of functions (\u03b1_n: n \u0404 \u03c9)\u2286 \u03c9^\u03c9 such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of \u03c9^\u03c9. We show that most of the usual definability results about the structure of countable subsets of \u03c9^\u03c9 have corresponding versions which hold about \u03c3-bounded subsets of \u03c9^\u03c9. For example, we show that every \u03a3_(2n+1^1 \u03c3-bounded subset of \u03c9^\u03c9 has a \u0394_(2n+1)^1 \"bound\" {\u03b1_m: m \u0404 \u03c9} and also that for any n \u2265 0 there are largest \u03c3-bounded \u03a0_(2n+1)^1 and \u03a3_(2n+2)^1 sets. We need here the axiom of projective determinacy if n \u2265 1. In order to study the notion of \u03c3-boundedness a simple game is devised which plays here a role similar to that of the standard ^*-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the ^*- and ^(**)-(or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of \u03c9^\u03c9 whose special cases include countability, being of the first category and \u03c3-boundedness and for which one can generalize all the main results of the present paper.", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1977-05", "volume": "229", "pages": "191-207" }, { "id": "authors:wa8kx-t2j47", "collection": "authors", "collection_id": "wa8kx-t2j47", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-103036161", "type": "article", "title": "Classifying Projective-like Hierarchies", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The starting point for the investigations reported in this paper is the following surprising result of Steel [St]:\nAssuming AD + DC let \u0413 be a collection of pointsets containing all open sets and closed under continuous preimages and countable intersections and unions. Then either \u0413 or \u0413 ( = the dual of \u0413) has the reduction property.\nThis result raises naturally the question if a similar phenomenon occurs with the stronger prewellordering property, perhaps under stronger closure assumptions on \u0413.", "issn": "0072-7466", "publisher": "Bulletin of the Greek Mathematical Society", "publication": "Bulletin of the Greek Mathematical Society", "publication_date": "1977", "series_number": "2", "volume": "18", "issue": "2", "pages": "254-275" }, { "id": "authors:8hmze-w5k35", "collection": "authors", "collection_id": "8hmze-w5k35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-111450945", "type": "article", "title": "\u220f^1_2 singletons and O^#", "author": [ { "family_name": "Harrington", "given_name": "Leo", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "A conjecture of Solovay states: Assuming that for every real \u0251, \u0251^# exists, the constructibility degrees of \u220f^1_2 singletons are wellordered and the successor steps in this wellordering are given by the sharps. In this paper we prove among others things that (assuming \u2200\u0251 (\u0251^# exists))\nfor every \u220f^1_2 singleton a either O^# is constructible from \u0251 or \u0251^# is constructible from O^#. From a relativized version of this result it follows that the constructibility degrees of O^#, O^(##), O^(###),... are the first \u03c9 constructibility degrees of sharps of \u220f^1_2 singletons.", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "1977", "series_number": "3", "volume": "95", "issue": "3", "pages": "167-171" }, { "id": "authors:836bv-kns72", "collection": "authors", "collection_id": "836bv-kns72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130624-085131170", "type": "article", "title": "On Monotone vs. Nonmonotone Induction", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Introduction. For definitions and notation in what follows, see [4] and [5].", "issn": "0273-0979", "publisher": "American Mathematical Society", "publication": "Bulletin of the American Mathematical Society", "publication_date": "1976-11", "series_number": "6", "volume": "82", "issue": "6", "pages": "888-890" }, { "id": "authors:225v3-pqd69", "collection": "authors", "collection_id": "225v3-pqd69", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-152738529", "type": "article", "title": "A Basis Result for \u03a3^0_3 Sets of Reals with an Application to Minimal Covers", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is shown that every \u2211^0_3 set of reals which contains reals of arbitrarily high Turing degree in the hyperarithmetic hierarchy contains reals of every Turing degree above the degree of Kleene's O. As an application it is shown that every Turing degree above the Turing degree of Kleene's O is a minimal cover.", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1975-12", "series_number": "2", "volume": "53", "issue": "2", "pages": "445-448" }, { "id": "authors:ayz3c-rjf46", "collection": "authors", "collection_id": "ayz3c-rjf46", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-073254315", "type": "article", "title": "A note on universal sets for classes of countable G_\u03b4'S", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "D. A.", "clpid": "Martin-D-A" } ], "abstract": "In a recent article [2] D. G. Larman and C. A. Rogers proved the following two results in Descriptive Set Theory (where R = the space of real numbers): (1) There is no analytic set in the plane R^2, which is universal for the countable closed subsets of R; (2) there is no Borel set in R^2, which is universal for the countable G_\u03b4 subsets of R. Recall that, if b is a class of subsets of a space X, a set U \u2286 X \u00d7 X is called universal for C if (\u0251) for each x \u0454 X, U_x = def {y : (x, y) U} \u0454 C, and (b) for each A \u0454 C there is an x such that A = U_x. (Larman and Rogers have also shown that in both cases coanalytic universal sets exist.)", "doi": "10.1112/S0025579300004484", "issn": "0025-5793", "publisher": "University College London", "publication": "Mathematika", "publication_date": "1975-06", "series_number": "1", "volume": "22", "issue": "1", "pages": "43-45" }, { "id": "authors:wvp16-k7081", "collection": "authors", "collection_id": "wvp16-k7081", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130524-135424900", "type": "article", "title": "On Characterizing Spector Classes", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study in this paper characterizations of various interesting classes of relations arising in recursion theory. We first determine which Spector classes on the structure of arithmetic arise from recursion in normal type 2 objects, giving a partial answer to a problem raised by Moschovakis [8], where the notion of Spector class was first\nessentially introduced. Our result here was independently discovered by S. G. Simpson (see [3]). We conclude our study of Spector classes by examining two simple relations between them and a natural hierarchy to which they give rise.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1975-03", "series_number": "1", "volume": "40", "issue": "1", "pages": "19-24" }, { "id": "authors:qjq3h-yjz89", "collection": "authors", "collection_id": "qjq3h-yjz89", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-132343731", "type": "article", "title": "The Theory of Countable Analytical Sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable \u220f_n^1 set of reals, C_n (this is also true for n even, replacing \u220f_n^1 by \u03a3_n^1 and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets C_n is then investigated in detail, the point of departure being the fact that each C_n is a set of \u0394_n^1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, \u03c9-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc.", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1975-02", "series_number": "2", "volume": "202", "issue": "2", "pages": "259-297" }, { "id": "authors:8q2ev-n7z83", "collection": "authors", "collection_id": "8q2ev-n7z83", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-115543154", "type": "article", "title": "Countable ordinals and the analytical hierarchy. I.", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The following results are proved, using the axiom of Projective Determinacy: (i) For n \u2ab4 1, every \u220f^1_(2n+1) set of countable ordinals contains a \u0394^1_(2n+1) ordinal, (ii) For n \u2ab4 1, the set of reals \u0394^1_(2n) in an ordinal is equal to the largest countable \u03a3^1_(2n) set and (iii) Every real is \u0394^1_n inside some transitive model of set theory if and only if n \u2ab4 4.", "issn": "0030-8730", "publisher": "Pacific Journal of Mathematics", "publication": "Pacific Journal of Mathematics", "publication_date": "1975", "series_number": "1", "volume": "60", "issue": "1", "pages": "223-227" }, { "id": "authors:hyf93-2sr27", "collection": "authors", "collection_id": "hyf93-2sr27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KECpjm75", "type": "article", "title": "Countable ordinals and the analytical hierarchy, I", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The following results are proved, using the axiom of Projective Determinacy: (i) For n \u2265 1, every II(1/2n+1) set of countable ordinals contains a \u0394(1/2n+1) ordinal, (ii) For n \u2265 1, the set of reals \u0394(1/2n) in an ordinal is equal to the largest countable \u03a3(1/2n) set and (iii) Every real is \u0394(1/n) inside some transitive model of set theory if and only if n \u2265 4.", "issn": "0030-8730", "publisher": "Pacific Journal of Mathematics", "publication": "Pacific Journal of Mathematics", "publication_date": "1975", "series_number": "1", "volume": "60", "issue": "1", "pages": "223-227" }, { "id": "authors:k61kb-3by18", "collection": "authors", "collection_id": "k61kb-3by18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-103426984", "type": "article", "title": "On Projective Ordinals", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study in this paper the projective ordinals \u03b4^1_n, where \u03b4^1_n = sup{\u03be: \u03be is the length of \u0251 \u0394^1_n prewellordering of the continuum}. These ordinals were introduced by Moschovakis in [8] to serve as a measure of the \"definable length\" of the continuum. We prove first in \u00a72 that projective determinacy implies \u03b4^1_n < \u03b4^1_n for all even n > 0 (the same result for odd n is due to Moschovakis). Next, in the context of full determinacy, we partly generalize (in \u00a73) the classical fact that \u03b4^1_1 \u2135_l and the result of Martin that \u03b4^1_3 = \u2135_(\u03c9 + 1) by proving that \u03b4^1_(n2+1) = \u03bb^+_(2n+1), where \u03bb_(2n+1) is a cardinal of cofinality \u03c9. Finally we discuss in \u00a74 the connection between the projective ordinals and Solovay's uniform indiscernibles. We prove among other things that \u2200\u03b1 (\u03b1^# exists) implies that every \u03b4^1_n with n \u2265 3 is a fixed point of the increasing enumeration of the uniform indiscernibles.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1974-06", "series_number": "2", "volume": "39", "issue": "2", "pages": "269-282" }, { "id": "authors:ymvrw-ces91", "collection": "authors", "collection_id": "ymvrw-ces91", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-122128969", "type": "article", "title": "Measure and category in effective descriptive set theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We are concerned in this paper with some definability aspects of the theory of measure and category on the continuum. Our objects of study are the projective subsets of the reals and their structure from a measure theoretic and topological point of view.", "doi": "10.1016/0003-4843(73)90012-0", "issn": "0003-4843", "publisher": "Elsevier", "publication": "Annals of Mathematical Logic", "publication_date": "1973-07", "series_number": "4", "volume": "5", "issue": "4", "pages": "337-384" }, { "id": "authors:ekx2r-k0d03", "collection": "authors", "collection_id": "ekx2r-k0d03", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-155104187", "type": "article", "title": "Two theorems about projective sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "In this paper we prove two (rather unrelated) theorems about projective sets. The first one asserts that subsets of \u2135_1 which are \u2211^1_2 in the codes are constructible; thus it extends the familiar theorem of Shoenfield that \u2211^1_2 subsets of \u03c9 are constructible. The second is concerned with largest countable \u2211^1_(2n) sets and establishes their existence under the hypothesis of Projective Determinacy and the assumption that there exist only countably many ordinal definable reals.", "doi": "10.1007/BF02764630", "issn": "0021-2172", "publisher": "Springer", "publication": "Israel Journal of Mathematics", "publication_date": "1972-12", "series_number": "4", "volume": "12", "issue": "4", "pages": "391-399" } ]