Simultaneous Embeddability of Two Partitions

We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability (weak, strong, and full embeddability) that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous embedding can be tested in linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201

On the complexity of the relations of isomorphism and bi-embeddability

Given an L_{\omega_1 \omega}-elementary class C, that is the collection of the countable models of some L_{\omega_1 \omega}-sentence, denote by \cong_C and \equiv_C the analytic equivalence relations of, respectively, isomorphism and bi-embeddability on C. Generalizing some questions of Louveau and Rosendal [LR05], in [FMR09] it was proposed the problem of determining which pairs of analytic equivalence relations (E,F) can be realized (up to Borel bireducibility) as pairs of the form (\cong_C,\equiv_C), C some L_{\omega_1 \omega}-elementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such problem: under very mild conditions on E and F, it is always possible to find such an L_{\omega_1 \omega}-elementary class C.Comment: 15 page

Invariantly universal analytic quasi-orders

We introduce the notion of an invariantly universal pair (S,E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in different areas of mathematics, when they are paired with natural equivalence relations.Comment: 31 pages, 1 figure, to appear in Transactions of the American Mathematical Societ

Superexpanders from group actions on compact manifolds

It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely embed into this Banach space. Combining this with non-embeddability results for warped cones by Nowak and Sawicki, which relate the non-embeddability of a warped cone to a spectral gap property of the underlying action, we provide new examples of expanders that do not coarsely embed into any Banach space with nontrivial type. Moreover, we prove that these expanders are not coarsely equivalent to a Lafforgue expander. In particular, we provide infinitely many coarsely distinct superexpanders that are not Lafforgue expanders. In addition, we prove a quasi-isometric rigidity result for warped cones.Comment: 16 pages, to appear in Geometriae Dedicat

Self-Referential Noise and the Synthesis of Three-Dimensional Space

Generalising results from Godel and Chaitin in mathematics suggests that self-referential systems contain intrinsic randomness. We argue that this is relevant to modelling the universe and show how three-dimensional space may arise from a non-geometric order-disorder model driven by self-referential noise.Comment: Figure labels correcte