Higher structures in higher-dimensional homological algebra

Homological algebra is a classical subject in mathematics that utilises algebraic methods to investigate ´obstructions´ of various kinds.

This proposal aims to investigate the higher structures that are present in an emerging variant of the theory called higher-dimensional homological algebra. Higher structures permit to refine classical invariants in order to encode subtle obstructions more faithfully.

The proposal is divided into four interlinked subprojects, whose topics range from generalisations of the classical reflection functors of Bernstein, Gelfand and Ponomarev to the symplectic geometry of symmetric products of surfaces and the study of enhancements of triangulated categories and higher-dimensional generalisations thereof.

The investigation of higher structures in *higher-dimensional* homological algebra is one of the novel aspects of this proposal.

Moreover, the systematic use of recently-developed methods from the theory of infinity-categories is also one of the novelties in our approach.

The results in this proposal are significant in that they will deepen our understanding of higher-dimensional homological algebra while also contributing to other subjects such as symplectic geometry and the general theory of triangulated categories.

The applicant´s research will be supported by a doctoral student, to be hired during the first year of funding, and two postdoctoral researches, to be hired in the first and third year of funding, respectively.