Isambard Goodbody

Project Title: Hochschild homology of differential graded algebras

Distinguishing objects is at the heart of modern mathematics; we want to reduce the problem of checking complicated properties to simple computations. The difficulty in checking properties is that one has to present objects. Different presentations of the same object can look very different, and it can be non-obvious when two presentations are of the same underlying object.

The solution to this difficulty is, in algebra and topology, the use of invariants, i.e. a way of transforming one object into a hopefully simpler one that doesn’t depend on the given presentation and is invariant under transformation. Some complete invariants entirely classify the objects up to a desired equivalence. Sometimes this is too optimistic and we must work with invariants that only preserve particular properties.

The proposed project is to study one such invariant, Hochschild homology, in the context of finite dimensional algebras. The central question, Han’s conjecture, is whether or not Hochschild homology is sufficient to determine the complexity of a finite dimensional algebra, where complexity is measured in terms of global dimension. We aim to use the added flexibility of differential graded algebras and employ techniques from homotopy theory to allow new computations.