Calabi-Yau algebras are very important in mathematics: as well as turning up naturally in representation theory and topology, they are noncommutative examples of Calabi-Yau varieties which play a central role in string theory. Lately, fractional Calabi-Yau algebras have risen in prominence. These are a more restrictive class of algebras that have very good homological properties. They connect abstract algebra, category theory, and noncommutative geometry. They can often be described using quivers (directed graphs) together with relations.
The aim of this project is to understand how fractional Calabi-Yau algebras arise in higher homological algebra. This involves constructing new examples of these algebras as well as studying their theoretical properties. Particular tools include the methods of abstract higher homological algebra, as well as methods inspired by mathematical physics. This project would suit a student who likes abstract algebra but is also interested in explicit computations. There are possibilities to use computer software in the research if that is of interest to the candidate.