Joint work with K. Baur was carried out towards the construction of a geometric model for skewed-gentle algebras in the first three months of the grant. However, in the meantime very similar results by other authors appeared on the arXiv (arXiv:2004.11136). We thus considered the class of string algebras instead, and obtained a geometric model for their module categories via partial triangulations of punctured surfaces, and a combinatorial description of their tau-tilting theory.
I started a new project with Emily Barnard, Emily Gunawan and Ralf Schiffler on a new class of representation theoretic objects, that of maximal almost rigid objects, By considering the geometric model constructed by K. Baur and I, we have a combinatorial description of maximal almost rigid objects over gentle algebras in terms of so-called permissible triangulations of a surface. This project is current work in progress.
In joint work with D. Pauksztello and D. Ploog, we established a bijection between simple-minded collections in the bounded derived category of an hereditary algebra sitting in the fundamental domain of the negative CY cluster category C and simpleminded systems in C. We also obtained a description of functorially finite hearts, which was key to obtain the bijection above.
Furthermore, we obtained a relationship between simple-minded systems and positive noncrossing partitions, generalising previous results by Iyama-Jin, Buan-Reiten-
Thomas and by me. These results are now published in Compositio Mathematica. This manuscript contains also an appendix containing joint work with D. Pauksztello
and A. Zvonareva on a reduction technique for simple-minded collections which does not involve Verdier localisations.
Partial results were also obtained on the mutation behaviour of simple-minded objects. In joint work with David Pauksztello, we showed that the left and right
mutations of w-simple-minded systems are again w-simple-minded systems, and that an almost complete w-simple-minded system has at least w complements. We also established an analogue result for simple-minded collections for which the extension closure is functorially finite. Moreover, we have a compatibility result between these mutations and HRS-tilting.
