Over the last two decades, higher homological algebra has emerged as a vibrant and extremely active area of contemporary mathematics. One of the big open questions in this field, is how the global data associated to an algebra determines existence of higher-dimensional structures. The project will build a bridge between higher homological algebra and the study of surface geometry, giving access to powerful geometric tools to solve this problem. Bringing together a diverse team of world-leading experts, the project aims to shape the future of higher homological algebra through significantly advancing the understanding of when and why higher homological phenomena arise.