Research interests

Mathematical logic; descriptive set theory; countable models; definable equivalence relations, especially those induced by the continuous actions of some Polish group; orbit equivalence.



Here are most of my papers from about 2000. Many of the papers written joint with other authors are missing. This list also does not include earlier papers.

 

Published papers

Classification and orbit equivalence relations (or the ps file; here there are a few diagrams which only come out reasonably when reading the ps file). This manuscript addresses the question of when an orbit equivalence relation allows countable models (considered up to isomorphism) as complete invariants. As well as developing a sufficient condition (``turbulence'') for non-classifiability and calculating the degree of classifiability for many concrete examples (measure preserving transformations up to isomorphism, compact metric spaces up to homeomorphism, homeomorphisms of various kinds of compact spaces, rank one torsion free abelian groups, and a few others) this work also provides a generalization of the Scott analysis to arbitrary Polish group actions. In order to be self contained, many of the recent results on Polish groups are recorded. (Published by the AMS in 2000.) (The file is largish. You can also buy a copy of this book from Amazon )

Vaught's conjecture on analytic sets (or the latex file). An algebraic characterization of which Polish groups fulfill the Vaught conjecture on any set arising as the continuous image of a Polish space: iff there is no closed subgroup that has the infinite symmetric group as a continuous homomorphic image. (Journal of the American Mathematical Society.)

Invariants for measure preserving transformations (or the latex file). The classification problem for measure preserving transformations is more difficult than for arbitrary countable structures (for instance countable linear orderings considered up to isomorphism, or countable groups considered up to isomorphism, countable graphs, countable fields, and so on). Some effort is also taken to analyze the specifically generalized discrete spectrum (or measure distal) ergodic transformations, in the sense of Furstenberg -- in particular showing non-classifiability by countable structures. (Fundamenta Mathematicae.)

The complexity of the classification of Riemann surfaces and complex manifolds (or the latex2e file). Illinois Journal of Mathematics. Joint with Alexander Kechris. Among other things, this paper shows that the isomorphism relation on complex surfaces is essentially countable but not essentially hyperfinite -- this is in contrast to the isomorphism relation on higher dimensional complex manifolds, which is not essentially countable.

A boundedness lemma for iterations (or the latex file). Using an observation regarding direct limits of coarse premice one can obtain exact bounds for prewellorderings uniformly definable from the reals and finitely many uniform indiscernibles over their relatively constructible universe. ( Journal of Symbolic Logic. The version posted here however contains an extended addendum on additional results which should be obtainable using the same technique.)

Conjugacy equivalence relation on subgroups (or the latex file). ( Fundamenta Mathematicae . Joint with Alex Andretta and Riccardo Camerlo.) If G is a finitely generated group containing a copy of the free (non-abelian group) then the conjugacy equivalence relation on its subgroups is universal.

Knight's model, its automorphism group, and characterizing the uncountable cardinals (or the latex file, or the ps file). There is a countable model whose Scott sentence characterizes the second uncountable cardinal, and there is a model similarly characterizing the first uncountable cardinal whose automorphism group is "divided" by the infinite symmetric group. The results here generalize an early paper by Julia Knight. This paper also gives a partial answer to a question about the actions of the automorphism group of Knight's model by forcing with the Frechet filter on the first uncountable ordinal. (Journal of Mathematical Logic.)

Cardinalities in the projective hierarchy (or the latex file, or the ps file). There are "more" sets at the second (third, fourth, etc) level of the projective hierarchy than there are at the first (second, third, etc).(Journal of Symbolic Logic)

A dichotomy theorem for turbulence (or the latex file, or the ps file). This paper gives the right dichotomy theorem for Borel orbit equivalence relations with respect to isomorphism on countable structures.(Journal of Symbolic Logic.)

Torsion free abelian groups (or the latex file). After an infuriatingly incorrect proof, and an unjustified earlier announcement, this contains what I can prove about countable TFA groups of infinite rank: The isomorphism relation is non-Borel (but I do not know whether it is above graph isomorphism). (Fundamenta Mathematicae)

Free continuous actions on zero-dimensional spaces. Joint with Mats Molberg. Various results about the dynamics of arbitrary groups acting freely on a zero-dimensional space. For instance: A topological version of the Adams counterexample; every countably infinite group acts freely and minimally on a non-homogeneous space. (Topology and its Applications.)

A converse to Dye's theorem. (or the latex file, or the ps file). A countable group is amenable if and only if up to OE it gives rise to exactly one countable ergodic measure preserving equivalence relation on a standard Borel probability space. There is a countable treeable equivalence which is neither hyperfinite nor universal. (Transactions of the AMS.)

Essentially countable equivalence relations. (or the latex file, or the ps file). They are not all bi-Borel reducible to a countable equivalence relation. (JSL).

The Furstenberg lemma characterizes amenability. (or the latex file, or the ps file). If an equivalence relation admits an equivariant assignment of probability measures for any cocycle into the homeomorphism group of a compact metric space, then it is amenable in the sense of Connes-Feldman-Weiss. A similar result holds for group actions being amenable in the sense of Zimmer. (PAMS.)

A note on counterexamples to the Vaught conjecture (or the postscript version). If there is a counterexample, then there is one with no model of size bigger than the first uncountable cardinal. (Notre Dame Journal of Formal Logic)

A lemma for cost attained (or the latex file, or the ps file). An ergodic equivalence attains cost n if and only if it is induced by a free action of the free group on n generators; more generally, treeable ergodic equivalence is always induced by the free action of some countable group. (This is in contrast to the recent theorem of Alexander Furman on certain kinds of restrictions of actions by lattices.) (Annals of Pure and Applied Logic.)

Subgroups of abelian Polish groups. (or the latex file, or the ps file). Following on from a paper by Farah and Solecki, this note shows that any uncountable abelian Polish group contains arbitrarily complicated Polishable subgroups. (Published in the conference volume for The Barcelona Year in Set Theory, 2003-4.)

Non-treeability for product group actions. (or the latex file, or the ps file). A lcsc group which is non-amenable and the product of two non-compact lcsc groups gives rise to a non-treeable orbit equivalence relation whenever it acts freely by measure preserving transformations on a standard Borel probability space. (Israel Journal of Mathematics).

An oscillation theorem for groups of isometries. (or the latex file, or the ps file). Given a non-trivial group of isometries on a space X we can find a bounded uniformly continuous function from X which continues to obtain high oscillation on the image of X under any isometric embedding arising in the closure of the group. There is a closely connected partition theorem for atomic models, stating that if the structure has non-trivial automorphism group then we can find a partition of some 2-type into two sets both of which are met in any elementary submodel. (To appear in GAFA.)

Randomness in effective descriptive set theory. Joint with Andre Nies. We considered what happens to some well known notions of randomness when recursively enumerable is replaced by its natural descriptive set theoretical counterpart. In this new context it turns out that low for random sets are all hyperarithmetical, but Kraft-Chaitin and Schnorr's theorem adapt. (Journal of the London Mathematical Society.)

Glimm-Effros for coanalytic equivalence relations (or the latex file, or the ps file.) A coanalytic equivalence relation either reduces the Vitali equivalence relation or admits bounded subsets of the countable ordinals as complete invariants. The note also gives a Harrington-Shelah type proof for a general form of Glimm-Effros. (To appear in the JSL.)

A selection theorem for treeable sets (or the latex file). If we have a Borel set in the plane with a tree structure on each of the sections, then the projection is Borel and there is a Borel selector. (To appear in PAMS.)

Borel equivalence relations which are highly unfree (or the latex file). There is a ergodic countable Borel equivalence relation on a standard Borel probability space which is not reducible to the free Borel action of a countable group on any non-null set. (To appear in the JSL.)

 

Preprints, notes, and preliminary reports

Non-smooth infinite dimensional group representation (or the latex file, or the ps file). It is known that the irreducible representations of a discrete group considered up unitary equivalence allow reals as complete invariants if and only if the group is abelian-by-finite. This brief note shows that when the group is not abelian-by-finite there is an immediate jump in the difficulty of the classification problem -- e.g. no Borel section meeting every equivalence class in a countable non-empty set.

A new zero-dimensional Polish group (or the latex file, or the ps file). This note presents a zero-dimensional Polish group that is generated by every open neighborhood of the identity.

A dichotomy theorem for being essentially countable. (or the latex file, or the ps file). Gives a dynamical analysis of when an orbit equivalence relation is reducible to an equivalence relation all of whose equivalence classes are countable.

A dichotomy for isomorphism (or the latex file, or the ps file). A Borel isomorphism relation is either essentially countable or as complicated as the countably infinite product of the Vitali equivalence relation. (RETRACTION: There is a problem in this proof...somewhere on like page 70 there is a mistake; spent a month figuring out how to fix it, then found another mistake a couple of pages later, which I am very unsure how to fix. The argument is certainly going to give something, and in many special cases can prove the result claimed, but right now I am too fed up with the thing to go back and figure out just how much can be saved.)

Relative ergodicity results for product group actions (or the latex file, or the ps file). Arguments to the effect that certain kinds of actions, such as Bernoulli shifts, are relatively ergodic with respect to actions by simpler product groups. The notes end with an elementary proof that there are incomparable countable Borel equivalence relations. (This is largely absorbed in to the book Kechris and I wrote together on product group actions, still the arguments here are a bit simpler than the final more general versions, so I am going to leave this proto-version up on line.)

Groups with HAP A countable group has the Haagerup approximation property if and only if the collection of mixing actions is dense in the space of all its measure preserving actions.

Rigidity and equivalence relations with infinitely many ends. Joint with Inessa Epstein. If a measure preserving equivalence relation on a standard Borel probability space admits a graphing with each equivalence class having infinitely many ends, then it is impossible to choose an end for each class in a Borel manner. Some applications to rigidity.

Independent axiomatizations. Joint with Yiannis Souldatos. Assuming Vaught's conjecture, any theory in countable infinitary logic of size continuum has an independent axiomatization.

An anticompleteness theorem. There is a complete, consistent, Borel theory having no Borel model. This is eventually going to appear as part of joint paper with Nies and Mountalban.

There are more coanalytic sets than Borel The effective cardinality of the collection of Borel sets is less than the effective cardinality of the collection of coanalytic sets. This is a consequence of a much stronger result in Andretta, Hjorth, Neeman, but the amusing proof here uses large cardinal ideas -- roughly, there are more definable classes of ordinals than sets.

Uniquely undefinable elements There is a model having a uniquely undefinable element over countably infinitary logic.

Two-generated groups are universal The isomorphism relation on groups generated by two of the elements is universal for countable Borel equvivalence relations.

Treeable equivalence relations There are continuum many incomparable treeable countable Borel equivalence relations.

Gaboriau-Popa reconsidered This brief note gives a proof of Gaboriau-Popa which does not use property (T).

The fine structure and Borel complexity of orbits This is an updated and considerably expanded version of a much older ``preliminary report". The basic project is to determine which Polish groups may admit a structural analysis of their orbits which has the profundity and force of the Scott analysis for the infinite symmetric group. To give this study concrete form, it is organized around a series of problems which are easily solved for the symmetric group using the Scott analysis, but remain unclear for general Polish groups. In rough terms, the paper presents an analysis which can do much, but not all, of the same work for general Polish groups. Some counterexamples and observations finally suggest that we should not expect a perfect solution for general Polish groups, but we may hope for that there will be cofinally many for which it is true.

 

Expository papers

Group actions and countable models. (or the ps file; or the tex file). A survey article presented at the ASL European meeting in Utrecht, 1999.

Notes from the Honolulu conference on abelian groups and modules (or the latex file, or the ps file). A brief survey article; a written version of the talk given at Honolulu.

Effective cardinals (or the latex file). An almost purely expository talk, based around my slides from Krakow, which provides a sketch of what a number of us are trying to do in this area.

Some notes (or the latex file, or the ps file) from a course I gave at Notre Dame. Gives a different proof of the Ulm-classifiability dichotomy theorem, just for the situation of countable models, but in a manner that is very elementary and requires no background in set theory.

An argument of due to Leo Harrington. (or the latex file). This presents Leo Harrington's unpublished solution to Friedman's problem 57*, concerning degree notions just below the hyperdegrees.

A survey of current and recent work on the theory of Borel equivalence relations. (or the latex file, and the bib file or the ps file.) This is for the upcoming and seemingly massive Handbook on Set Theory, editored by Matt Foreman and Aki Kanamori.

Countable Borel equivalence relations, Borel reducibility, and orbit equivalence. Based on four talks given at Kobe, as part of a minicourse for ALC 10. Giving brief proofs of results regarding orbit equivalence and why set theorists have taken an interest in the area. A lot of proofs are given, or at least sketched; written from the point of view of a set theorist who would like to find out more about amenability, property (T), etc.

Vienna notes on effective descriptive set theory, Spector-Gandy, Barwise compactness, and Gandy-Harrington forcing.

 

Links

Some papers on related themes are also at John Clemens' homepage, Ilijas Farah's homepage, Matt Foreman's homepage, Damien Gaboriau's homepage, Su Gao's homepage, Alexander Kechris' homepage, Nicolas Monod's homepage, Sorin Popa's homepage, Christian Rosendal's homepage, Slawek Solecki's homepage, Simon Thomas' homepage, and the set theory and its neighbours homepage.

Click here to get a listing of Hjorth's papers from the AMS MathSciNet with links to Mathematical Reviews. (Note: You or your institution must have a valid MathSciNet subscription.)

 




The work listed above has received partial support from NSF grants DMS 0140503, DMS 99-70403, and DMS 96-22977 as well as a fellowship from the Sloan foundation.