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#### Descripción Breve de Actividad Realizada

Review of journal articles:

Wild Cantor actions:

This paper was the first incursion of the author in the realm of Cantor actions (see Section 5 of the report). The main contribution of this paper was the construction of several new families of examples of minimal, equicontinuous Cantor actions that exhibit interesting properties in their discriminant subgroups. The examples are constructed as actions on rooted trees and the acting groups are countable subgroups of the product or of the wreath product of groups. There are also some applications of the results to the study of attractors of dynamical systems and of minimal sets of foliations

Chaotic Delone sets.

This paper introduced a generalization of cut-and-project tilings to the hyperbolic setting. One takes a cocompact lattice of PSL(2,R), a geodesic in the hyperbolic plane and some suitable parameter r>0. Every time the geodesic passes at distance less than r>0 from a point in the lattice, we add the point in the geodesic closest to the lattice. In this way, we obtain a Delone set; it is proved that the natural dynamical system associated to this object is chaotic, and that chaos is generic in a suitable configuration space of Delone sets.

Coarse distinguishability of graphs with symmetric growth:

Let X be a connected, locally finite graph with symmetric growth. This paper proves the existence of a coloring φ : X → {0, 1} and some R ∈ N such that every automorphism f preserving φ is R-close to the identity map; this can be seen as a coarse geometric version of symmetry breaking. As an application, it proves that the infinite motion conjecture of Tom Tucker is true for graphs where at least one vertex stabilizer satisfies a certain dynamical condition.

Genericity of chaos for colored graphs:

To each colored graph one can associate its closure in the universal space of isomorphism classes of pointed colored graphs, and this subspace can be regarded as a generalized subshift. In a sense, this is a graph version of some of the ideas present in the first paper. Two definitions for chaotic colored graphs are introduced, one of them analogous to Devaney’s, showing the equivalence of our two novel definitions of chaos and proving their topological genericity.

Molino’s description and foliated homogeneity:

We introduce an analogue of the discriminant group (see Section 5) to this setting. The triviality of this compact group has a clear dynamical interpretation. We also give an example where the projection of the classical Molino's description is not a principal bundle, and another example of positive topological codimension where the foliated homogeneity cannot be checked by only comparing pairs of leaves.

Bounded geometry and leaves:

The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space. The article combined some of the ideas present in the previous one with some new tools from analysis and Riemannian geometry.

This article has achieved a Dimensions index of 4.21, meaning that it has been cited 4.21 times more often than the average of the field. It has been cited on twice on “Geometry & Topology” and once on “International Mathematical Research Notes”, two very reputable journals.

A universal Riemannian foliated space.

This was the first paper by the applicant, in collaboration with his supervisor at the time (Álvarez López) and Candel from CSUN, USA. The problem that inspired this work was the question by Sondow asking which manifolds can be realized as leaves of foliations on compact manifolds. The strategy was to define a suitable universal space of pointed Riemannian manifolds endowed with the topology of smooth convergence of manifolds, and then finding applications of this space to several areas (including the aforementioned realization problem).

The results of this paper proved useful in the long term because of the ability of this universal space to serve as the foundation for studying unimodular random manifolds. The article achieved a Dimensions index of 6, meaning that it has been cited 6 times more often than the average of the field. It has been cited on the Duke Mathematical Journal, one of the best mathematical journals in the world, and on Annals de l’Institute Fourier and Geometry&Topology, two very reputable journals.

B.1. Breve descripción del Trabajo de Fin de Grado (TFG) y puntuación obtenida El trabajo de fin de grado llevó el título "Espacios fibrados vectoriales". Comprendió desde su definicion y propiedades básicas hasta teoría más avanzada, relacionada con la K teoría y teoría de categorías. Se estudiaron también ejemplos concretos. La nota obtenida fue un 9.0B.2. Breve descripción del Trabajo de Fin de Máster (TFM) y puntuación obtenida El rótulo del trabajo de fin de máster fue "Acciones distales", donde se trató principalmente el teorema de Furstenberg sobre la estructura de las acciones distales como torre de límites. La nota obtenida fue un 10.