In short, my work focuses on combinatorial models of higher categories. Higher categories are a generalisation of spaces, adding a notion of “direction” to the latter: While in a space any path can be travelled along in two directions, in higher categories this need not be the case. This is relevant for the description of irreversible processes. In my PhD thesis I build a fully algebraic and computer implementable model of higher categories by observing that higher categories are locally described by certain manifolds embedded in directed (Euclidean) space. Higher categories can be a foundation for all of mathematics. They encompass many interesting (and possibly all feasible) modes of composition in mathematics, vastly extending upon the linear mode of composition of functions in Set Theory. In future research I would like to follow this line of thought further, ultimately arriving at a type theory for higher categories. Somewhat speculatively, this could be useful in many parts of physics, in particular quantum physics and topological field theories, which I would like to explore as well.
A more detailed research statement is available upon request.
A more detailed CV is available upon request.