Research

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

Recently, I have been working on problems in spectral geometry. This area was popularized by a famous question posed by Kac in the 1960s: "Can one hear the shape of a drum?" In more general terms, Kac's question asked whether the geometry of a space (the shape of a drum) is completely determined by the Laplace spectrum (the sound of a drum). While there are now many examples proving that the answer is "no," there is no denying that the Laplace spectrum is closely related to the geometry of spaces. For instance, the Laplace spectrum determines the lengths of closed geodesics on Riemann surfaces. For general Riemannian manifolds, the Laplace spectrum determines the dimension, volume, and total curvature. Additionally, there are numerous bounds of eigenvalues in terms of various geometric quantities in the literature. Spectral geometry is not only interesting on its own, but it also has numerous exciting applications in other problems in mathematics.

I mainly study spectral geometry through the lens of symmetry, which is a powerful tool in geometry as it provides insightful examples or interesting testing grounds for general conjectures.  In my research, I have utilized symmetries in both ways. Another exciting recent development for me is exploring spectral geometry in connection with an unconventional notion of symmetry known as singular Riemannian foliations (SRFs).

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)
  to appear in Geometry and Topology, arxiv

• Spectral Multiplicity and Nodal Sets for Generic Torus-Invariant Metrics,
  (with Donato Cianci, Chris Judge, and Craig Sutton)
  International Mathematics Research Notices (2024), no. 3, 2192–2218 article

• Geometric Structures and the Laplace Spectrum, Part II,
  (with Benjamin Schmidt and Craig Sutton)
  Transactions of American Mathematical Society, 374 (2021), 8483-8530, article

• Geometric Structures and the Laplace Spectrum,
  (with Benjamin Schmidt and Craig Sutton)
  Annales de l'Institute Fourier, 74(2024) no. 2 pp.867-914,  article

• Curvature Free Rigidity for Higher Rank Three-manifolds,
  Indiana University Mathematics Journal, 67, (2018), no.6, 1597-1623, article

• Manifolds With Many Hyperbolic Planes,
  (with Benjamin Schmidt)
  Differential Geometry and its Applications, 52 (2017), 121-126, article

• Real Projective Space with All Geodesics Closed,
  (with Benjamin Schmidt)
  Geometric and Functional Analysis, 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