I define myself as an algebraic topologist. I am interested in understanding groups, mainly finite ones, as groups of automorphisms of non-associative algebras/rings, smooth manifolds, or topolological spaces. So my research focus on the interactions between Group Theory, Ring Theory, Differential Geometry, and Homotopy Theory, and it follows different lines:
Homotopy theory of loop spaces: By homotopy theory of loop spaces I mean "homotopy group theory", "p-compact groups", or "p-local finite groups", and related stuff as cohomology of groups. This implies that sometimes I have to work with honest groups. I develop this research as part of the Barcelona Algebraic Topology Group.
Homotopy invariants and self homotopy equivalences: This is a more classical branch of Algebraic Topology in which I am also interested. My research is nowadays mainly focus in understanding the group of self homotopy equivalences of a space. This research is developed jointly with Cristina Costoya as part of the Xunta de Galicia research project "Representación Homotópica de Grupos" (EM2013/016).
Groups of diffeomorphisms: This is closely related with the previous research line. I am interested in constructing manifolds with "very few diffeomorphisms" or describing a given group of diffeomorphisms as automorphisms of a vector field. I develop this research jointly with Francisco Javier Turiel at Málaga.
All these approaches to Homotopy Theory are included in the research FEDER-MCI grants MTM2013-41768-P, MTM2016-78647-P, PID2020-118753GB-I00, and PID2023-149804NB-I00.
I am a proud member of the following mathematical research networks:
RET: Red Española de Topología (Spanish Topology Network). It has a facebook page!
IBG: Red Ibérica de Teoría de Grupos (Iberian Network of Group Theory).
NcAlg: Red Temática de Álgebra no Conmutativa (Noncommutative Algebra Thematic Network).