// research
Working at the intersection of differential geometry, conformal geometry, quantum gravity, and cosmology.
// overview
I study the deep mathematical structure of spacetime — from the geometry of its boundaries to the quantum fluctuations at its smallest scales.
My research spans conformal geometry, general relativity, quantum gravity, and cosmology. On the geometric side, I develop a systematic theory of conformal hypersurface invariants — canonical extrinsic curvatures that govern when conformally compact manifolds are related to Poincaré-Einstein metrics — with applications to Willmore energies, Q-curvatures, and holographic boundary problems.
On the quantum gravity and cosmology side, I have studied curvature fluctuations in discrete models of quantum spacetime and explored holographic dark energy scenarios, including constraints on exotic cosmological futures such as the "long freeze." These threads are unified by a broader interest in understanding the deep mathematical structure of spacetime, from its microscopic quantum origins to its large-scale cosmological evolution.
// research interests
// selected publications
We establish the existence of a new symmetric trace-free tensor conformal invariant of hypersurface embeddings in even-dimensional conformal manifolds, completing the family of conformal fundamental forms. The object has important links to global problems and to the Poincaré–Einstein boundary value problem.
We study the nonlinear Dirichlet-to-Neumann map for the Poincaré–Einstein filling problem. For even-dimensional manifolds, the range of this non-local map is described in terms of a rank-two tensor along the boundary, proportional to the variation of renormalized volume.
We introduce a new geometric structure — potential Carroll structures — on three-manifolds equipped with a preferred direction, and determine the conditions relating them to special Carrollian manifolds (SCMs), with implications for null hypersurface geometry and holography.
Understanding the microscopic behavior of spacetime is critical for developing a theory of quantum gravity. We analyze curvature fluctuations in a discrete, baby model of quantum spacetime, probing how classical geometry emerges from quantum structure.
We study the geometric invariant theory of null hypersurfaces, developing a Carrollian hypersurface calculus for intrinsic and extrinsic differential invariants. Motivated by black hole thermodynamics, we argue that a connection with torsion is the most natural object to study Carrollian manifolds.
We generalize holographic methods to higher-codimension submanifolds, extracting higher-order local invariants of both Riemannian and conformal embeddings. New behavior arises in the higher-codimension case, giving rise to invariants that obstruct the order-by-order construction of unit defining maps.
We explicitly compute the extrinsic Paneitz operator and apply it to obtain an extrinsically-coupled Q-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds, and generalized Willmore energies in all even dimensions.
We construct a sequence of conformally invariant higher fundamental forms and show that their vanishing is a necessary and sufficient condition for a conformally compact metric to be conformally related to an asymptotically Poincaré-Einstein metric.