// 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, and everything in between.
My research spans conformal geometry, general relativity, quantum gravity, and cosmology. On the geometric side, I developed a systematic theory of conformal hypersurface invariants: canonical extrinsic curvatures that govern when conformally compact manifolds are related to Poincaré-Einstein metrics. These invariants have 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
// publications & eprints
We introduce potential Carroll structures on three-manifolds and determine conditions relating them to special Carrollian manifolds, with implications for null hypersurface geometry and holography.
We show that Einstein gravity coupled to Yang-Mills theory implies a conformally invariant Yang-Mills equation on the conformal boundary, with applications to holographic gauge theories.
We establish a new symmetric trace-free tensor conformal invariant of hypersurface embeddings in even-dimensional conformal manifolds, completing the family of conformal fundamental forms, with connections to the Poincaré–Einstein boundary value problem.
We prove the general non-existence of higher-order conformal fundamental forms when the embedded hypersurface is odd-dimensional, completing the characterization of conformal fundamental forms.
We demonstrate the consistency of a breathing universe scenario, contributing to the growing literature on alternative cosmological models beyond standard ΛCDM.
Using conformal geometry, we find necessary conditions for generic solutions to asymptotically approach the Kerr–de Sitter metric, constraining the admissible form of the free data on conformal infinity.
We explicitly determine all shear-free null hypersurfaces embedded in Einstein spacetimes and show each corresponds to a Carrollian structure with a unique pair of Ehresmann connection and affine connection.
We analyze quantum curvature fluctuations in a baby quantum gravity model, deriving implications for the cosmological constant from a single-plaquette discretization of spacetime.
We provide a natural generalization to submanifolds of the holographic method used to extract higher-order local invariants of both Riemannian and conformal embeddings, with qualitatively new behavior in higher codimension.
We study holographic dark energy models admitting a "long freeze" in which the scale factor evolves to a constant, analyzing the viability of this exotic cosmological scenario.
We study the geometric invariant theory of null hypersurfaces, developing a Carrollian hypersurface calculus for intrinsic and extrinsic differential invariants of black hole horizons.
We explicitly compute the extrinsic Paneitz operator and apply it to obtain an extrinsically-coupled Q-curvature for embedded four-manifolds and generalized Willmore energies in all even dimensions.
We study the nonlinear Dirichlet-to-Neumann map for the Poincaré–Einstein filling problem, constructing conformally invariant Dirichlet-to-Neumann hypersurface invariants.
We construct conformally invariant higher fundamental forms and show their vanishing is necessary and sufficient for a conformally compact metric to be conformally related to an asymptotically Poincaré–Einstein metric.