Currently I am working on the Hochschild homology of curved category algebras. Hochschild homology is an invariant of associative rings (not necessarily commutative). To any category we can associate a ring called the category algebra, and the curved element is an element of the center of this ring. In the case of toric varieties, curved category algebras are related to Landau-Ginzburg models and matrix factorizations. However, they are more concrete to work with, especially when it comes to the Hochschild homology.
My previous research project was about the Koszulity of the endomorphism algebra of the tilting object when we have a full strong exceptional collection. I generalized some existing Koszulity results, which establish a connection between Koszulity and Cohen-Macaulay property, in a categorical framework. Then I applied them to the cases that we have a full strong exceptional collection. Here is a preprint of my recent paper.