The classical point set continuum is unphysical in that it requires the simultaneous determination of a non-denumerable infinity of distances. This is contrary to the principles of quantum mechanics.

LC's work in quantum gravity began with a series of papers on categorical state sums and topological quantum field theory. The most recent work in this series is on a new 2-categorical model.

In these models, smooth manifolds are replaced by a special class of simplicial complexes. These complexes have more singular topology than a manifold.

In recent papers, LC has studied possible applications of the analysis of such singular configurations to early universe phenomenology and particle physics.

More recently, he has been exploring the possibility of using the categorical state sums to define an abstract homotopy theory on a higher model category. This is an approach to interpreting the state sums as approximations to a more fundamental theory.

The abandonment of an absolute physical point set, to be replaced by a relational topology in which regions have relative point sets which they can detect in other regions is meant to be a fundamental approach to the problem of the infinities. It is also mathematically quite natural within the context of the theory of schemes, topoi and stacks. Hopefully, a physical principal and an established branch of Mathematics fuse.

LC's work appears with various coauthors on gr-qc and Q-alg.