Research

My research interests revolve around the study of non-Hausdorff manifolds, and their potential application to problems of topology change in quantum gravity. Here’s a brief summary for the curious mathematician:

Topologically, non-Hausdorff manifolds can always be realised as a union of Hausdorff ones — if you are feeling Zorn-y then you can take a cover by open charts and extend it into a cover by maximal open Hausdorff submanifolds, and if you are feeling particularly fancy then you can represent non-Hausdorff manifolds as a colimit in the category TOP. The latter is a nice approach, because it informs us of how to carry other geometric data of interest over into the non-Hausdorff setting. Locally-defined geometry passes over without issue, but global features are pathological — things like Stokes’ Theorem, integral results such as the Gauss-Bonnet Theorem, and the general cohomology of non-Hausdorff manifolds become tricky without access to arbitrarily-existent partitions of unity. But, these global constructions are indeed possible and now well-understood.

Once all of the geometric structure is defined and properly characterized, we can start to ask bold questions like “what would a non-Hausdorff spacetime look like?” The short answer is something like below, where here the level sets are displayed on the right hand side:

My main interest is in building a theory of physics based off of these weird guys. Mostly I care about the inclusion of these spaces into a path integral for gravity in Lorentzian signature, since there is always a question of which topologies to sum over in the path integral, and it appears as though objects like the above ought to be preferred over conventional topology-changing spacetimes such as the almost-Lorentzian cobordisms. In the past, I have studied the potential inclusion of these non-Hausdorff spacetimes within a path integral for gravity in 2d (which can be found here).

These days, I am mostly thinking about categorical extensions of QFTs into the non-Hausdorff setting. The idea sounds difficult, but it’s really not: if we restrict our attention to particular non-Hausdorff colimits in some nice-enough category of globally-hyperbolic spacetimes that already admit a locally-covariant QFT, then the real question boils down to the correct choice of non-Hausdorff test functions, and the correct categorical construction to impose on all the algebras we already have access to. After this there is the fun prospect of constructing Hadamard states on a non-Hausdorff spacetime — with causal structures like those pictured above, wavefront sets presumably behave in a strange way. But, armed with appropriate categorical constructions, perhaps it is still possible to glue compatible Hadamard states defined on all of the Hausdorff pieces of the manifold.

Dare I say it, I may even venture into some non-non-Hausdorff geometry some time soon — I am curious about the Index Theory of almost-Lorentzian cobordisms, since I refined a proof of the Lorentzian Gauss-Bonnet Theorem semi-recently. It would be interesting to see how much of the discrete approach is “smoothable” and generalizes to a Lorentzian version of the Atiyah-Singer Index Theorem. If you are interested in this then please reach out!