Quivers, exact categories, and A-infinity algebras

The research project aims to significantly advance our knowledge on exact categories by developing a theory of quiver and relations for them, using homological and homotopical algebra like A-infinity algebras, corings, and semialgebras.Exact categories were introduced by Quillen in 1972 as a framework for algebraic K-theory.

They are a generalisation of abelian categories that keeps a notion of short exact sequences, but drops general existence of (co)kernels and the first isomorphism theorem.Exact categories are abundant throughout mathematics, including Banach spaces in functional analysis,vector bundles and Cohen-Macaulay modules in algebraic geometry, categories of modules with a Verma flag in representation theory, but even monomorphism categories in topological data analysis.An indispensable tool in the study of abelian categories has been the notion of a quiver and relations.

This notion has been a key ingredient in the theory of cluster algebras, e.g. leading to a proof of the Zamolodchikov periodicity conjecture in theoretical physics, in tilting theory, leading to Kontsevich´s homological mirror symmetry, and in the theory of Hall algebras, which is used to study Donaldson-Thomas invariants for Calabi-Yau threefolds.The applicant´s joint work on existence, resp. uniqueness of exact Borel subalgebras for quasi-hereditary algebras with Koenig and Ovsienko, resp.

Miemietz, can be seen as an important instance for the theory we aim to develop and expand on in general.