Group Theory is a subfield of Algebra. It’s aim is to understand symmetries of complicated mathematical objects and to relate them to other features of these objects, such as geometric properties. The initial object can, for example, be a difficult number-theoretical equation that needs to be solved. The symmetries then form an algebraic structure called a group. Groups themselves often have additional geometric or number-theoretical features in a natural way. Some groups also result from the asymptotic study of infinite sequences of finite, but ever-increasing complicated structures. Link

A fundamental result from logic states that in many situations there are idealized versions of the mathematical objects that you are actually interested in. For example, if you are interested in very small or very large numbers, you can use this result to work with infinitely small or infinitely large numbers instead, which often leads to simplifications. This method can be applied to almost all objects that occur in Algebra. Model Theory consists of systematically investigating what is possible in this way and using this approach to solve algebraic problems.  Link

Algebra and Geometry are closely intertwined fields in modern mathematical research. Geometric and topological questions can often be answered with algebraic methods using so-called invariants. In our work we pay special attention to the family of L2 invariants. The latter reflect algebraic, measure-theoretic and asymptotic properties of various geometric objects. Another subject of our research is the phenomenon of profinite rigidity of groups. Groups describe symmetries. For infinite groups, one wonders whether finite approximations of these symmetries already determine the shape of the group.  Link