The three main lines of research of the group are Group Theory, Topology and Applications.
In group theory, our research group works in the fields of geometric group theory, and finite and profinite groups.
The field of geometric group theory emerged from Gromov?s insight that even mathematical objects such as groups, which are defined completely in algebraic terms, can be profitably viewed as geometric objects and studied with geometric techniques. Contemporary geometric group theory has broadened its scope considerably, but retains this basic philosophy of reformulating in geometric terms problems from diverse areas of mathematics and then solving them with a variety of tools. The growing list of areas where this general approach has been successful includes low-dimensional topology, the theory of manifolds, algebraic topology, complex dynamics, combinatorial group theory, algebra, logic, the study of various classical families of groups, Riemannian geometry and representation theory.
Our second main research line in group theory deals with finite and profinite groups. The study of these two classes of groups complements each other in many situations: results from finite group theory can be used to prove facts about profinite groups, and conversely profinite methods can be applied to obtain results about finite groups via the use of inverse limits. This connection has been succesfully exploited to prove a number of important results in group theory in this century. No doubt, finite group theory reached its peak in the past century with the classification of finite simple groups in the 1980s. However, at the other end of the spectrum, there are still many interesting open problems regarding finite p-groups, and we address several of them in this project. This naturally leads to the study of similar questions in pro-p groups, which can be often restricted to p-adic analytic groups. We consider a wide variety of topics in this line, ranging from character theory and representation theory to cohomological properties. We also deal with word problems, which are enjoying a revival in the last few years, with the proof of very strong results, such as the Ore conjecture.
Since their formation as branches of mathematics, group theory and topology intertwine and go hand in hand: one way of understanding the structure of topological spaces is via their group of symmetries, and a fruitful way of understanding the algebraic structure of a group is by making it act on a topological space that one understands.
In the area of topology, our group works on pointfree topology. Pointfree topology is a well established area of study of what is usually called categorical topology. It is a modern algebraic approach to topology on a constructive foundation, using the tools of category theory and lattice theory, with nice and interesting ramifications in logic and computation (because it can be dealt in an impredicative constructive setting, e.g. in a topos). One of the main uses of pointfree topology is to reformulate definitions and results from classical mathematics, replacing the standard point-set notions with the corresponding point-free notions (points and continuous functions are derived notions; instead basic open sets and their covering and approximation relations are taken as fundamental). This idea has been pursued in mathematics for almost 30 years, and is quite common within computer science and topos theory.
Our research is in pure mathematics, nevertheless, our group has outreach towards real life applications. Among them are applications to medicine, in particular to the design of vaccines and the design of DNA codes. On the other hand, along with our external collaborators, we seek to implement new cryptographic protocols based on group theory.
- Geometric and combinatorial group theory
- Finite and profinite groups
- Ordered structures and topology