The research is developed in several directions within the scope of topology (general and categorical topology, convergence spaces and quasi-uniform spaces) and its relationships with the infinite group theory (topological groups and their duality, Zariski topology of infinite groups, and its applications). We present three concrete examples below.

The notion of topological entropy was introduced for continuous maps of compact spaces, and later it was generalised for other areas. Recently, the topological entropy of actions of amenable groups has been intensely studied. The algebraic entropy through group endomorphisms is a dual notion with respect to the topological one, and recently it was studied by many authors in various forms. In particular, the Yuzvinski Formula allows the expression of the famous Lehmer's Problem of the number theory in terms of values of algebraic entropy. Therefore, the topological entropy and the algebraic entropy are both studied, as well as their mutual relationship and that with other numerical invariants such as the scale function of Willis, within the scope of locally compact abelian groups, both for individual endomorphisms and in the case of actions of amenable semi-groups. Also, the language of algebraic entropy allows us to extend the notion of growth from the classical scope of the geometric group theory, in which the famous Milnor's Problem was solved by Gromov and by Grigorchuk, to the case of group endomorphisms. The interest, therefore, is focused on understanding the new notion of growth by group endomorphisms following the path outlined by classical results. Within the geometric group theory, the coarse spaces, in the sense meant by John Roe, are studied, too.

The theory of characterised subgroups of topological abelian groups was originated by the study of the dynamical behaviour of the rotations of the unit circle T, and it is linked with problems of Diophantine approximations (uniformly distributed successions, Kronecker successions, continued fractions) and with the theory of the sets of convergence of trigonometric series in harmonic analysis (Dirichlet sets, Arbault sets, etc.). Another motive underpinning this theory comes from the notion of topological torsion, introduced in the study of the structure of locally compact abelian groups. Several authors achieved many results in this field, in connection with the number theory, the automata theory etc.; on the other hand, many problems are still open, not least the description of all the characterised subgroups of T. Therefore, the characterised subgroups of topological locally compact abelian groups are studied, with a particular interest in the fundamental case of T, also in connection with Dirichlet sets.

According to Mackey's Theorem, each locally convex metrisable vector space admits the Mackey topology. Under this respect, we study the locally quasi-convex abelian groups admitting a Mackey topology, with particular attention being paid to the metrisable case and to the group's algebraic structure.