RESEARCH GROUP

GEOMETRY

I am an algebraist by training and a topologist/geometer in spirit, which means I work in group theory. That is, my research concerns properties of spaces and of their symmetries (such as reflections). The fun part is that the term "space" can be interpreted quite broadly, from a hairy ball to a comic-book-like multiverse known as Bruhat--Tits building, from fundamental geometric shapes to model spaces describing the motion of a robot.

In this area one can investigate a wide variety of problems. For example, regarding the space we could ask whether there exist serious obstructions in it (such as holes of various dimensions), whether it is curved somehow, what happens when we move "forever" (e.g., "towards infinity"), or whether we can describe what happens over time with an object moving in the given space under certain rules.

As for the symmetries themselves, we might want to know for instance how many there are, whether we can describe them in a simple fashion, how they transform objects in the given space, whether there are points left unchanged by the symmetries, or about the relationship between given sets of symmetries of two spaces (possibly even the same!).

In more technical terms, my work so far has been mostly in combinatorial/geometric group theory and related topics from topology. I had/have particular interest in presentations, (cohomological) finiteness conditions, Reidemeister classes, and algorithmic questions. As such, I am very fond of geometric, topological, homological, combinatorial, and algorithmic aspects of groups and spaces on which they act.

Groups that I like include (but are not limited to) linear groups (algebraic and Lie groups, their (S-)arithmetic counterparts, Coxeter groups, ...), R. Thompson's groups and their relatives, and locally compact (including profinite) groups. In the topological realm I also enjoy things like knots and links (and more generally spatial graphs) and questions about low-dimensional spaces.

As a former member of the group of Theory of Compuation in Brasília, I am also interested in formal methods in mathematics and proof assistants. I recent times I have also been puzzled by Costas arrays.