The main research activities focus on
Representation theory of quivers and algebras as well as its applications in other areas of mathematics. Our research leverages, in particular, methods from homological algebra and higher category theory.
Structure and representation theory for vertex operator algebras. These algebraic structures are foundational for conformal field theory and string theory in theoretical physics.
Solving systems of polynomial equations using combinations of algebraic geometry and numerical linear algebra. This research has connections to the project "Polynomial equations in geometry and computer vision".
Computational group theory, in particular investigation of maximal symmetry groups of hyperbolic space using algorithmic methods.
Classifying spaces of compact Lie groups. This involves p-compact groups and fusion systems but also group cohomology and representation theory. The work often involves the use of computer algebra.
The theory of profinite groups, in particular pro-p groups, and includes groups acting on rooted trees such as branch groups. This research has connections to the theory of finite p-groups, to Hausdorff dimensions and fractal structures, and has also links to computer science and dynamics.
Algorithms for constructing canonical bases of ideals (Gröbner bases) and subalgebras (SAGBI bases) and the usage of such bases to solve geometric and algebraic problems.