Research

I am not doing any active mathematical research at the moment.

My research was mainly about representation theory of finite-dimensional algebras. In particular I liked to study algebras that are interesting from the point of view of higher (mostly 2-) dimensional Auslander-Reiten theory. A key concept in this theory is that of cluster tilting, which was originally introduced to categorify the cluster algebras of Fomin and Zelevinsky. The theory of cluster algebras is itself very rich, and has connections to other subjects such as Teichmüller theory.

It makes sense to study endomorphism algebras of cluster tilting objects, and these turn out to be very special algebras called Jacobian algebras. They can be described by the datum of a quiver with potential or QP, i.e. a quiver with a given formal linear combination of oriented cycles. Moreover, self-injective Jacobian algebra appear in 2-dimensional AR theory as analogues of preprojective algebras of Dynkin quivers.

My research was about constructing and studying (self-injective) Jacobian algebras coming from several sources, such as Postnikov diagrams, skew group algebras and surfaces. Some broad problems I looked at (and said something about) are:

• Consider the QPs constructed as in Baur-King-Marsh, but for arbitrary strand permutation. Find the self-injective ones. Can this class of QPs be described intrinsically?
• Generalise the construction of Amiot-Plamondon to arbitrary surfaces with orbifold points. Save what we can save on the cluster side.
• Generalise both constructions to strand diagrams on surfaces, and on surfaces with orbifold points. As a first case, study the disk with one orbifold point.

I am also usually interested in any kind of algebraic problems with a combinatorial flavour, and I like working with other people a lot (though at the moment I do not have much time). So if you have ideas you would like to share, feel free to contact me.