My research
My research
I am interested in applying categorical polynomial functors (like in work by David Spivak) to describe how power grid components (like generators, inverters, and transmission lines) interact in oscillatory systems. These functors could model the emergent oscillations that occur in power grids when they undergo reconfiguration, faults, or stress conditions.
If you are interested in this too or have ideas or concerns, please reach out to me.
I am interested in a wide variety of topics, however the projects below are the ones I currently am most active in.
Tangent categories generalize multiple approaches to geometry (differential geometry, Synthetic differential geometry, algebraic geometry). A central object of study in tangent categories are differential bundles, genereralizing vector bundles and allowing for the notion of connections. The goal of this project is to find a functorial characterization of differential bundles, generalize it to tangent infinity categories and understand some basic properties.
A characterization of differential bundles using functors from a category of free N-modules is going to appear on the arXiv soon.
Like the previous project generalizes vector bundles, this project aims to generalize principal G-bundles. In differential geometry and topology a principal bundle is a space that is locally isomorphic to a base space times a group in such a way that the transition between charts is given by the group multiplication. In order to generalize principal bundles the setup of join restriction categories is used as they have a natural way for two spaces to be locally isomorphic.
The behaviour of principal G-bundles in tangent join restriction is particular interesting, not just because it is the situation differential geometry is in, but also because the differential structure from the tangent categories interact in surprising ways with the local structure of a join restriction category.
Additionally we observe some interesting behaviour of group objects in tangent categories that lead to an external Lie-Algebra generalizing the Lie-algebra of a Lie-group.
The paper is available on the arXiv now: https://arxiv.org/abs/2509.18410
In quantum mechanics one often considers superchannels, transformations between quantum channels. Understanding the higher categorical structure of superchannels will make it possible to solve long-standing problem on the concrete implementation of quantum superchannels, which in turn will allow us to develop new quantum communication protocols and optimize quantum computation algorithms. The key structure is the Choi isomorphism that identifies transformations between channels with elements in a space of transformations and thereby provides a closed monoidals structure.
For this we are developing a graphical calculus for symmectric monoidal closed categories. This graphical calculus should graphically identify A->(B->C) and (A x B)->C and thereby provide an easy to use tool for scholars in physics and computer science.
Abelian functor calculus provides a way to approximate functors with chain complexes in an Abelian category as their target. In work with Kristine Bauer, Ea E and Jason Parker we showed that for the bicategory AbCatCh of Abelian categories, functors into chain complexes (with Kleisli-style composition) and natural transformations up to natural chain homotopy equivalences, the homotopy category Ho(AbCatCh) becomes a Cartesian Differential Category. This amounts to proving that the degree 1 approximation of the functor calculus fulfills a chain rule analogous to the chain rule in classical calculus.