Research

My research interest is in Riemannian geometry, including spectral geometry, homogeneous spaces, and dynamics of geodesics.

In recent years, I have been working on problems in spectral geometry. This area is not only attractive on its own but also has many connections to other problems in differential geometry. My current work explores how symmetries are encoded in the eigenvalues and eigenfunctions of the Laplace operator and other differential or pseudo-differential operators. 

Besides spectral geometry, I have also worked on geometric rigidity theory. Locally symmetric spaces form an essential class of manifolds in Riemannian geometry. Geometric rigidity theory aims to characterize these spaces in terms of their geometric and dynamical properties by proving rigidity theorems. Geometric rigidity theory has a long history, tracing back to the celebrated Mostow's rigidity theorem. I recommend this survey by Prof. Spatzier for an introduction.

• A Weyl's Law for Singular Riemannian Foliations with Applications to Invariant Theory, with Ricardo Mendes and Marco Radeschi, submitted, arxiv

• Spectral Multiplicity and Nodal Sets for Generic Torus-Invariant Metrics, with Donato Cianci, Chris Judge, and Craig Sutton, Int. Math. Res. Not. IMRN (2024), no. 3, 2192–2218 article

• Geometric Structures and the Laplace Spectrum, Part II, with Benjamin Schmidt and Craig Sutton, Trans. Amer. Math. Soc. 374 (2021), 8483-8530, article

• Geometric Structures and the Laplace Spectrum, with Benjamin Schmidt and Craig Sutton, Ann. Inst. Fourier 74(2024) no. 2 pp.867-914,  article

• Curvature Free Rigidity for Higher Rank Three-manifolds, Indiana Univ. Math. J., 67, (2018), no.6, 1597-1623, article

• Manifolds With Many Hyperbolic Planes, with Benjamin Schmidt, Differential Geom. Appl. 52 (2017), 121-126, article

• Real Projective Space with All Geodesics Closed, with Benjamin Schmidt, Geom. Func. Anal. 27, (2017), no. 3, 631-636, article

• Isometric Torus Actions and the Eigenfunctions of the Dirichlet-to-Neumann Operator, with Chris Judge and Craig Sutton.

• Generic Irreducibility of Eigenspaces for Non-free Torus Actions, with Chris Judge and Craig Sutton.

Here are some videos of the talks I gave online:

• Spectral Multiplicity and Nodal Domains of Torus-invariant Metrics (November 2023, Spectral Geometry in the Clouds) YouTube Video

• Geometric Structures and the Laplace Spectrum (January 2021, Virtual Seminars on Geometry with Symmetries) YouTube Video