Related papers: The Complexity of Connectedness Relations on Polis…
A framework to handle tree decompositions of the components of a Borel graph in a Borel fashion is introduced, along the lines of Tserunyan's Stallings Theorem for equivalence relations arXiv:1805.09506. This setting leads to a notion of…
We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ER s of a family which includes Polish actions of S_\infty, the group of all permutations of N, and is closed…
We study local connectedness, local accessibility and finite connectedness at the boundary, in relation to the compactness of the Mazurkiewicz completion of a bounded domain in a metric space. For countably connected planar domains we…
For $f \colon [0,1] \rar \real^{+}$, consider the relation $\mathbf{E}_{f}$ on $[0,1]^{\omega}$ defined by $(x_{n}) \mathbf{E}_{f} (y_{n}) \Leftrightarrow \sum_{n < \omega} f(|y_{n} - x_{n}|) < \infty.$ We study the Borel reducibility of…
Chain total double complexes with reductive differentials for non-abelian simplexes with associated spaces are considered. It is conjectured that corresponding relative cohomology is equivalent to the coset space of vanishing over…
In this paper we address the question: How many pairwise non-isomorphic extremely amenable groups are there which are separable metrizable or even Polish? We show that there are continuum many such groups. In fact we construct continuum…
A topological setting is defined to study the complexities of the relation of equivalence of embeddings (or "position") of a Banach space into another and of the relation of isomorphism of complex structures on a real Banach space. The…
Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the…
We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to…
We give, for each level of complexity L, a Hurewicz-like characterization of the Borel subsets with countable sections of a product of two Polish spaces that cannot become in L by changing the two Polish topologies.
We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…
We characterise the embeddability of simply connected locally 3-connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski's characterisation of graph planarity, by excluded minors. This answers questions of…
We indicate a way of distinguishing between structures, for which, we call two structures distinguishable. Roughly, being distinguishable means that they differ in the number of realizations each gives for some formula. Being…
We study the descriptive complexity of sets of points defined by placing restrictions on statistical behaviour of their orbits in dynamical systems on Polish spaces. A particular examples of such sets are the set of generic points of a…
Coskey, Hamkins, and Miller [CHM12] proposed two possible analogues of the class of countable Borel equivalence relations in the setting of computable reducibility of equivalence relations on the computably enumerable (c.e.) sets. The first…
We study analysis on the cone of discrete Radon measures over a locally compact Polish space $X$. We discuss probability measures on the cone and the corresponding correlation measures and correlation functions on the sub-cone of finite…
Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If…
We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra…
We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, resp. pseudonorms, on the countable infinite-dimensional rational vector space. We provide an exhaustive…
Let $E\subseteq F$ and $E'\subseteq F'$ be Borel equivalence relations on the standard Borel spaces $X$ and $Y$, respectively. The pair $(E,F)$ is simultaneously Borel reducible to the pair $(E',F')$ if there is a Borel function $f:X\to Y$…