The present page contains brief information, which may be useful to students, who approach me for a PhD, or for an MSc, but also to colleagues who would like to know about my recent interests of research.

1. Probabilistic methods in group theory.

I am mainly interested in the theory of the commuting elements in algebraic structures (and corresponding generalizations). There are some classical contributions of Paul Erdős, who wrote a series of influential papers in the 1960s. I like to study both the finite case and the infinite case, dealing with compact groups since they have naturally the Haar measure. It has been shown a parallel approach in terms of character theory, so there are relations with the representation theory of groups (or in general of algebraic structures) and probabilities on groups. It is also possible to find connections with the theory of the random walks, because it is possible to look at problems of commutativity from the perspective of the dynamical systems and this gives many relations with measure theory and mathematical physics.

2. Nilpotent groups and related Lie algebras.

I like to investigate the classical theory of finite p-groups (p prime) and the corresponding infinite version in the context of profinite groups. One of my main interests deals with the classification of p-groups via Schur multiplier. This object can be treated both with homological machineries and with appropriate generators and relations, so its theory fits very well many categories and that of Lie algebras shows interesting analogies which I explore systematically. I have recently found (with surprise) that the theory of the Schur multipliers has applications in mathematical models, where Heisenberg groups and Heisenberg algebras are involved. Finally I study more theoretical aspects of structure of nilpotent groups, involving techniques and ideas of the so-called theory of the formations of Gaschütz.

3. Locally compact groups.

The general theory of locally compact groups is fundamental in several areas of pure and applied mathematics such as Abstract Harmonic Analysis, Topology and Algebra. Beginning from certain problems regarding profinite groups, it is quite natural to work by analogy and generalize notions, ideas and techniques to the case of locally compact groups. I am mainly interested to study locally compact groups with methods of combinatorial topology, homology and graph theory, beginning from problems, or curiosities, that I note when I study profinite groups. In particular, I wrote a monograph on a family of locally compact groups which present significant restrictions on the lattice of closed subgroups. These groups were known since long time, but a systematic approach was absent in the literature for locally compact groups.

4. Isoperimetric inequalities in algebraic structures.

I had the chance to study this interesting field of research in connection with classical results of Spectral Graph Theory due to F. Chung and S.T. Yau of almost twenty years ago. The idea is to adapt some technical notions of differential geometry, like the notion of isoperimetric dimension, or the notion of Cheeger constant, to the context of metric spaces. Graphs fits very well this context. The general framework requires variations of the Bochner formula (and this is quite subtle but possible), so that each geometric structure has its own peculiarity and one is involved directly in studying geometric analysis (very often on graphs).

 5. Geometric Group Theory.

I have mainly studied two problems. The first is the so called qsf property, due to seminal papers of V. Poenaru, A. Casson and J. Stallings. These works involve an application of certain techniques of differential geometry to those finitely presented groups which can be naturally associated to a manifold via fundamental groups. The main point is to show a conjecture of Stallings for which there are no non-qsf finitely presented groups. Its proof is at a final stage, after several classes of finitely presente groups have been shown to be qsf. This conjecture would give an example of a geometric property which is nontrivial and somehow plays an anomalous role in the well-accepted perspective of the geometric group theory ``a la Gromov''. The second problem deals with the so called Dehn functions and, more generally, with the properties of the word metric on finitely presented groups. This field involves technquies of combinatorics and differential geometry.

6. Applications of topology to dynamical systems.

I found surprising to justify a sophisticated mathematical model, which involves the dynamics of neutron stars, via pure considerations of topology. In fact I am recently interested in applications of the topology in complex dynamical systems. There are a series of important open problems about the analytic integration of governing equations in mathematical physics and the use of symmetries (and of the topology) seems to give significant solutions.