Nicholas Touikan

Research interests

I am a geometric group theorist, that is to say a person who studies groups as geometric objects. In particular, I study finitely generated groups, which are discrete objects, like the vertices of a graph. One could say that the geometry of a finite set of points is sort of boring. I would have to agree, and this is why I study infinite groups.

Even though at a small scale these groups don’t really have interesting geometry, at a large scale they may have interesting features such as large scale negative curvature. Much of my research deals with exploiting aspects of large scale negative curvature to solve computational problems in groups and describe their algebraic structure.

One may also realise a group as the fundamental group of a topological space and, by performing operation on topological spaces, one performs analogous operations on fundamental groups. For example, the common doughnut has an infinite cyclic, or Z, fundamental group.

The operation of joining two donuts produces an “infinity doughnut” whose fundamental group is the amalgamated product Z*Z. This gives me another way to use geometry, specifically low dimensional topology, to study groups: by performing operations on spaces related to a group, it becomes possible to gain insights into the group itself which were not immediately apparent from a purely algebraic viewpoint.

Outside of group theory, my research interests intersect mathematical logic and computer science… but I kind of like everything!