Random Subgroups of Rationals

This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup (G,+) of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of (G,+); second, what learnability properties can one extract from G and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of (G,+) coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for G with respect to any generating sequence for G is not even semi-decidable, one can build a generating sequence beta such that the word problem for G with respect to beta is co-recursively enumerable (assuming that the set of generators of G is limit-recursive). In regard to the second question, it is proven that there is a generating sequence beta for G such that every non-trivial finitely generated subgroup of G is recursively enumerable and the class of all such subgroups of G is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of G is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of G cannot be syntactically identified in the limit with respect to any generating sequence for G. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups

Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups

We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana

Group actions on 1-manifolds: a list of very concrete open questions

This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear panorama on the subject arises from the lecture.Comment: 21 pages, 2 figure

Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper