16 research outputs found
Games orbits play and obstructions to Borel reducibility
We introduce a new, game-theoretic approach to anti-classification results
for orbit equivalence relations. Within this framework, we give a short
conceptual proof of Hjorth's turbulence theorem. We also introduce a new
dynamical criterion providing an obstruction to classification by orbits of CLI
groups. We apply this criterion to the relation of equality of countable sets
of reals, and the relations of unitary conjugacy of unitary and selfadjoint
operators on the separable infinite-dimensional Hilbert space.Comment: 13 pages. Final version, to appear in Groups, Geometry, and Dynamic
Structures and dynamics
Our results are divided in three independent chapters.
In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same applies to the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: "For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M)." From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15].
In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16].
In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16]
Incompleteness Theorems for Observables in General Relativity
The quest for complete observables in general relativity has been a
longstanding open problem. We employ methods from descriptive set theory to
show that no complete observable on rich enough collections of spacetimes is
Borel definable. In fact, we show that it is consistent with the
Zermelo-Fraenkel and Dependent Choice axioms that no complete observable for
rich collections of spacetimes exists whatsoever. In a nutshell, this implies
that the Problem of Observables is to 'analysis' what the Delian Problem was to
'straightedge and compass'. Our results remain true even after restricting the
space of solutions to vacuum solutions. In other words, the issue can be traced
to the presence of local degrees of freedom. We discuss the next steps in a
research program that aims to further uncover this novel connection between
theoretical physics and descriptive set theory.Comment: 5 pages, 2 figure